Andrew Walker Problems

Exercise Show that $\{1+i,1-i\}$ form a linearly independent set of vectors in $\mathbb {C}$, viewed as a vector space over $\mathbb {R}$.

Proof Recall that the set of vectors $\{v_{1},\ldots ,v_{n}\}$ in a vector space $V$ (over a field $\mathbb {F}$) are said to be linearly independent if whenever $c_{1},\ldots ,c_{n}$ are scalars in $\mathbb {F}$ such that $c_{1}v_{1}+\cdots +c_{n}v_{n}=0,$ then $c_{1}=\cdots =c_{n}=0$. So for this problem, since we’re considering the complex numbers $\mathbb {C}$ as a vector space over $\mathbb {R}$, we must show that whenever $c_{1},c_{2}\in \mathbb {R}$ and $c_{1}(1+i)+c_{2}(1-i)=0,$ then $c_{1}=c_{2}=0$. Rearranging the above equation, we obtain $(c_{1}+c_{2})+(c_{1}-c_{2})i=0.$ Now, a complex number is equal to $0$ if and only if its real and imaginary parts are both $0$. So in this case, we conclude that $c_{1}+c_{2}=0{\text{ and }}c_{1}-c_{2}=0.$ This implies $c_{1}=c_{2}$, so that $c_{1}+c_{2}=2c_{2}=0$, which yields $c_{1}=c_{2}=0$. Thus we conclude the vectors $1+i,1-i$ are linearly independent in $\mathbb {C}$ (over $\mathbb {R}$).

Exercise Show that $\{1+i,1-i\}$ form a linearly independent set of vectors in $\mathbb {C}$, viewed as a vector space over $\mathbb {R}$.

Proof Recall that a set of vectors $\{v_{1},\ldots ,v_{n}\}$ in a vector space $V$ (over a field $\mathbb {F}$) is said to be ${\textbf {linearlydependent}}$ if they are not linearly independent. More concretely, these vectors are linearly dependent if we can find scalars $c_{1},\ldots ,c_{n}\in \mathbb {F}$ not all equal to zero such that $c_{1}v_{1}+\cdots +c_{n}v_{n}=0.$

So for this problem, to show that $1+i$ and $1-i$ are not linearly dependent over $\mathbb {C}$, all we need to do is exhibit two complex scalars $c_{1}$ and $c_{2}$ that are not both zero such that $c_{1}(1+i)+c_{2}(1-i)=0.$ There are many choices for $c_{1}$ and $c_{2}$, but one such example is $c_{1}=i$ and $c_{2}=1$.

Exercise Let $V$ be a vector space over a field $\mathbb {F}$. If $\{v_{1},v_{2},v_{3},v_{4}\}\subseteq V$ are a linearly independent set of vectors, then show that $\{v_{1}-v_{2},v_{2}-v_{3},v_{3}-v_{4},v_{4}\}$ also form a linearly independent set of vectors in $V$.

Proof Recall that the set of vectors $\{w_{1},\ldots ,w_{n}\}$ in a vector space $V$ (over a field $\mathbb {F}$) are said to be linearly independent if whenever $c_{1},\ldots ,c_{n}$ are scalars in $\mathbb {F}$ such that $c_{1}w_{1}+\cdots +c_{n}w_{n}=0,$ then $c_{1}=\cdots =c_{n}=0$.

Exercise So for this problem, we must show that whenever $c_{1},c_{2},c_{3},c_{4}\in \mathbb {F}$ and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_{1}(v_{1} - v_{2}) + c_{2}(v_{2} - v_{3}) + c_{3}(v_{3} - v_{4}) + c_{4}v_{4} = 0,} we have that $c_{1}=c_{2}=c_{3}=c_{4}=0.$ After rearranging terms in the above equation, we have that $c_{1}v_{1}+(c_{2}-c_{1})v_{2}+(c_{3}-c_{2})v_{3}+(c_{4}-c_{3})v_{4}=0.$ Now since the vectors $\{v_{1},v_{2},v_{3},v_{4}\}$ are linearly independent in $V$ by assumption, we have that

$c_{1}=0$

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_{2} - c_{1} = 0 }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_{3} - c_{2} = 0 }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_{4} - c_{3} = 0.}

In other words, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_{1} = c_{2} = c_{3} = c_{4} = 0} , so that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{v_{1} - v_{2}, v_{2} - v_{3}, v_{3} - v_{4},v_{4}\}} form a linearly independent set as desired.

Exercise Prove that a vector space Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} over a field Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{F}} is infinite-dimensional if and only if there is a sequence Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_{1},v_{2},\ldots} in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_{1},\ldots,v_{m}} is linearly independent for every Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m \in \mathbb{N}} .

Proof Recall that a vector space Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} is said to be finite dimensional if it is spanned by a finite list of vectors $w_{1},\ldots ,w_{m}\in V.$ In other words, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} has finite dimension if every vector in $V$ may be written as a linear combination of some list of vectors $w_{1},\ldots ,w_{m}\in V$. On the other hand, a vector space $V$ is infinite dimensional if it is not finite dimensional, i.e., $V$ cannot be spanned by a finite list of vectors. Now before we proceed in the proof, we will need the following fact:

Lemma Suppose $V$ is a vector space over a field $\mathbb {F}$, and $v_{1},\ldots ,v_{n}$ are vectors that span $V$. If $w_{1},\ldots ,w_{m}$ in $V$ are linearly independent, then $m\leq n$.

We are ready now to proceed with the proof:
’$\Rightarrow$’: Suppose that $V$ is an infinite dimensional vector space. Then, in particular, $V\neq 0$, so that there is some $v_{1}\neq 0$ in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} . Then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_{1}} is a linearly independent vector in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} . By way of induction now, suppose that for some Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k \geq 1} , we have produced vectors $v_{1},\ldots ,v_{k}\in V$ such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_{1},\ldots,v_{k}} are linearly independent. Since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} is infinite-dimensional, it cannot be spanned by the (finite!) list of vectors Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_{1},\ldots,v_{k}} . Thus we have that there is some Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_{k+1} \in V} such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_{k+1} \neq c_{1}v_{1} + \cdots + c_{k}v_{k}, \text{ for any }c_{1},\ldots, c_{k} \in \mathbb{F}.} We claim that now that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_{1},\ldots,v_{k},v_{k+1}} form a linearly independent set in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} . To see this, suppose that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_{1}v_{1} + \cdots + a_{k}v_{k} + a_{k+1}v_{k+1} = 0 \text{ for some } a_{1},\ldots,a_{k},a_{k+1} \in \mathbb{F}.} Now if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_{k+1} \neq 0} , then we may re-write the above equation as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_{k+1} = \Big( \frac{-a_{1}}{a_{k+1}} \Big)v_{1} + \cdots + \Big( \frac{-a_{k}}{a_{k+1}} \Big)v_{k},} contradicting the fact that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_{k+1}} is not in the span of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_{1},\ldots,v_{k}} . So we conclude Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_{k+1} = 0} , and thus we have that $a_{1}v_{1}+\cdots +a_{k}v_{k}+a_{k+1}v_{k+1}=a_{1}v_{1}+\cdots +a_{k}v_{k}+(0)v_{k+1}$ $=a_{1}v_{1}+\cdots +a_{k}v_{k}=0.$ Now by induction hypothesis, since $v_{1},\ldots ,v_{k}$ are linearly independent, we must have Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_{1},\ldots,a_{k}} are all zero. Thus we’ve shown that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_{1},\ldots, v_{k},v_{k+1}} also form a linearly independent set, completing the induction. Thus we have constructed a sequence of vectors Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{v_{k}\}^{\infty}_{k =1}} in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} so that $v_{1},\ldots ,v_{m}$ is linearly independent for each Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m \in \mathbb{N}} .

’Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Leftarrow} ’: On the other hand, suppose that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} contains a sequence of vectors Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{v_{k}\}^{\infty}_{k =1}} so that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_{1},\ldots, v_{m}} is linearly independent for each $m\in \mathbb {N}$. By way of contradiction, let’s suppose Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} is not infinite dimensional, i.e. is finite dimensional. Then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} can be spanned by a finite list of vectors Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w_{1},\ldots, w_{n} \in V} .
Now, since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} contains a linearly independent set of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} .

Exercise Suppose that $U$ and $W$ are subspaces of a vector space $V$. Prove that $U\cup W$ is a subspace of $V$ if and only if $U\subseteq W$ or $W\subseteq U$.

Proof Recall that a subset $A$ of a vector space $V$ is a subspace of $V$ if $A$ itself is a vector space with the same addition and scalar multiplication operations as $V$.
’$\Rightarrow$’: Instead of proving that $U\cup W$ is a subspace of $V$ implies $U\subseteq W$ or $W\subseteq U$, we’ll show the contrapositive of this statement. That is, if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U \not\subseteq W} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W \not\subseteq U} , then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U \cup W} is not a subspace of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} . So suppose there is some Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in U} that is not in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W} , and likewise that there is some Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y \in W} that is not in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U} . We claim that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x + y \notin U \cup W} . For if it were, then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x+y} would lie in either Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U} or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W} . If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x + y \in U} , then since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U} is a subspace, this would imply Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = (x+y) - x \in U,} contradicting our choice of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} . Likewise, if $x+y\in W$, this would yield $x\in W$, which is again a contradiction. So we conclude that $x+y\notin U\cup W$, and thus $U\cup W$ fails to be closed under addition, so cannot be a subspace of $V$.
’$\Leftarrow$’: Suppose now that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U \subseteq W} or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W \subseteq U} . Then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U \cup W} is equal to either Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W} or $U$ respectively, which, by assumption are subspaces of $V$.

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Before we begin, we will need the following notation: for an arbitrary non-empty set $X$, let $\mathbb {R} ^{X}$ denote the set of all functions $f\colon X\to \mathbb {R}$. Then $\mathbb {R} ^{X}$ is always a vector space, with addition and scalar multiplication defined pointwise.

Exercise Let $b\in \mathbb {R}$ and consider the set $W={\Big \{}f\in \mathbb {R} ^{[0,1]}\colon f{\text{ is continuous and}}\int _{0}^{1}f(x)dx=b{\Big \}}.$ Show that $W$ is a subspace of $\mathbb {R} ^{[0,1]}$ if and only if $b=0$.

Proof Recall that a subset $A$ of a vector space $V$ is a subspace of $V$ if $A$ itself is a vector space with the same addition and scalar multiplication operations as $V$. There is a very convenient test that determines if $A$ is a subspace of $V$, sometimes called the subspace test. It says the following:

(Subspace Test) Suppose that $A\subseteq V$, where $V$ is a vector space over a field $\mathbb {F}$. Then $A$ is a subspace of $V$ if and only if the following conditions are met:

1. $0_{V}\in A$,
2. $c\in \mathbb {F} ,v\in A\Rightarrow cv\in A$
3. $v,w\in A\Rightarrow v+w\in A$.

We are now ready to proceed with the proof:

’$\Rightarrow$’: Suppose $W$ is a subspace of $\mathbb {R} ^{[0,1]}$. Then by condition $(1)$ of the subspace test, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W} contains the zero vector of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^{[0,1]}} , which is just the function that maps Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0} for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in [0,1]} . We will write this zero vector as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \textbf{0}} . Now since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \textbf{0} \in W} , by definition of being in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W} , we must have that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int^{1}_{0} \textbf{0}(x)dx = b.} On the other hand, when we actually integrate Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \textbf{0}} , we find the integral must be zero. Thus $b=0$ as desired.
’$\Leftarrow$’: Say $b=0$. We will show that $W$ is a subspace of $V$ by showing that it passes all three conditions of the subspace test above. For condition (1), just note that by our previous remark, $\int _{0}^{1}{\textbf {0}}(x)dx=0=b$, and since ${\textbf {0}}$ is continuous, we have that ${\textbf {0}}\in W$. For condition (2), suppose that $c\in \mathbb {R}$ and $f\in W$. We must show that $cf\in W$. Since a continuous function multiplied by a constant is still continuous, $cf$ is still a continuous function. Now, $\int _{0}^{1}(cf)(x)dx=\int _{0}^{1}c[f(x)]dx=c\int _{0}^{1}f(x)dx=c(0)=0,$ so that we conclude Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle cf \in W} . Lastly, for condition (3), we must show that if $f,g\in W$, then $f+g\in W$. The addition of two continuous functions is always continuous, so that $f+g$ is continuous. Now since $f,g\in W$, we have that $\int _{0}^{1}(f+g)(x)dx=\int _{0}^{1}[f(x)+g(x)]dx=\int _{0}^{1}f(x)dx+\int _{0}^{1}g(x)dx=0+0=0,$ so that $f+g\in W$, and thus $W$ satisfies all three conditions of the subspace test.

Exercise Prove or give a counterexample to the following statement: If $U_{1},U_{2},W$ are subspaces of a vector space $V$ with $V=U_{1}\oplus W{\text{ and }}V=U_{2}\oplus W,$ then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U_{1} = U_{2}} .

Proof Let’s think about what it means for two subspaces $A,B$ of a vector space $C$ to satisfy Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C = A \oplus B} . This means that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A \cap B = \{0_{C}\}} and that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C = A + B} . In other words, for any Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c \in C} , we may write Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} uniquely in the form Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c = a + b} , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \in A} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b \in B} .
It turns out the statement of the problem is false, so that we must provide a counterexample to this statement: Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V = \mathbb{R}^{2}} and consider its subspaces (one should check that they actually form subspaces first): Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W = \{ (x,0) \in \mathbb{R}^{2} \colon x \in \mathbb{R} \} <br /> U_{1} = \{ (0,y) \in \mathbb{R}^{2} \colon y \in \mathbb{R} \} <br /> U_{2} = \{ (z,z) \in \mathbb{R}^{2} \colon z \in \mathbb{R} \} }

Then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U_{1}} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U_{2}} are not the same subspaces of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} , so that all we need to check is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U_{1} \oplus W = V = U_{2} \oplus W.} Suppose that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x,y) \in V} . Then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x,y) = (x,0) + (0,y)} , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x,0) \in W} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (0,y) \in U_{1}} , so that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V = W + U_{1}} . Now if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a,b) \in W \cap U_{1}} , then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a,b) = (x,0) \in W} for some Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in \mathbb{R}} , and thus Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b = 0} . Likewise, $(a,b)=(0,y)\in W$ for some $y\in \mathbb {R}$, so that $a=0$. This shows $W\cap U_{1}=\{(0,0)\}$, and hence $V=U_{1}\oplus W$.
Now again say $(x,y)\in V$. Then $(x,y)=(x-y,0)+(y,y)$, where $(x-y,0)\in W$ and $(y,y)\in U_{2}$ so that $V=W+U_{2}$. Now suppose $(a,b)\in W\cap U_{2}$. Then $(a,b)=(x,0)\in W$ for some $x\in \mathbb {R}$, so that $b=0$. Likewise, $(a,b)=(a,0)=(z,z)\in U_{2}$ for some $z\in \mathbb {R}$, thus $z=0=a$, so that we conclude $W\cap U_{2}=\{(0,0)\}$, and thus $V=U_{2}\oplus W$.