031 Review Part 2, Problem 4

Suppose  $T$  is a linear transformation given by the formula

$T{\Bigg (}{\begin{bmatrix}x_{1}\\x_{2}\\x_{3}\\\end{bmatrix}}{\Bigg )}={\begin{bmatrix}5x_{1}-2.5x_{2}+10x_{3}\\-x_{1}+0.5x_{2}-2x_{3}\end{bmatrix}}$

(a) Find the standard matrix for  $T.$

(b) Let  ${\vec {u}}=7{\vec {e_{1}}}-4{\vec {e_{2}}}.$  Find  $T({\vec {u}}).$

(c) Is  ${\begin{bmatrix}-1\\3\end{bmatrix}}$  in the range of  $T?$  Explain.

Foundations:
1. The standard matrix of a linear transformation  $T:\mathbb {R} ^{n}\rightarrow \mathbb {R} ^{m}$  is given by
${\begin{bmatrix}T({\vec {e_{1}}})&T({\vec {e_{2}}})&\cdots &T({\vec {e_{n}}})\end{bmatrix}}$
where  $\{e_{1},e_{2},\ldots ,e_{n}\}$  is the standard basis of  $\mathbb {R} ^{n}.$
2. A vector  ${\vec {x}}$  is in the image of  $T$  if there exists  ${\vec {x}}$  such that
$T({\vec {x}})={\vec {v}}.$

Solution:

(a)

Step 1:
Notice, we have
$T({\vec {e_{1}}})={\begin{bmatrix}5\\-1\end{bmatrix}},T({\vec {e_{2}}})={\begin{bmatrix}-2.5\\0.5\end{bmatrix}},T({\vec {e_{3}}})={\begin{bmatrix}10\\-2\end{bmatrix}}.$

Step 2:
So, the standard matrix of  $T$  is
$[T]={\begin{bmatrix}5&-2.5&10\\-1&0.5&-2\end{bmatrix}}$

(b)

Step 1:
Since  $T$  is a linear transformation, we know
${\begin{array}{rcl}\displaystyle {T({\vec {u}})}&=&\displaystyle {T(7{\vec {e_{1}}}-4{\vec {e_{2}}})}\\&&\\&=&\displaystyle {T(7{\vec {e_{1}}})-T(4{\vec {e_{2}}})}\\&&\\&=&\displaystyle {7T({\vec {e_{1}}})-4T({\vec {e_{2}}}).}\end{array}}$

Step 2:
Now, we 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 \begin{array}{rcl} \displaystyle{T(\vec{u})} & = & \displaystyle{7\begin{bmatrix} 5 \\ -1 \end{bmatrix}-4\begin{bmatrix} -2.5 \\ 0.5 \end{bmatrix}}\\ &&\\ & = & \displaystyle{\begin{bmatrix} 35 \\ -7 \end{bmatrix}+\begin{bmatrix} 10 \\ -2 \end{bmatrix}}\\ &&\\ & = & \displaystyle{\begin{bmatrix} 45 \\ -9 \end{bmatrix}.} \end{array}}

(c)

Step 1:
To answer this question, we augment the standard matrix 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 T}   with this vector and row reduce this matrix.
So, we have the matrix
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 \left[\begin{array}{ccc|c} 5 & -2.5 & 10 & -1\\ -1 & 0.5 & -2 & 3 \end{array}\right].}

Step 2:
Now, row reducing this matrix, we 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 \begin{array}{rcl} \displaystyle{\left[\begin{array}{ccc|c} 5 & -2.5 & 10 & -1\\ -1 & 0.5 & -2 & 3 \end{array}\right]} & \sim & \displaystyle{\left[\begin{array}{ccc|c} 5 & -2.5 & 10 & -1\\ -5 & 2.5 & -10 & 15 \end{array}\right]}\\ &&\\ & \sim & \displaystyle{\left[\begin{array}{ccc|c} 5 & -2.5 & 10 & -1\\ 0 & 0 & 0 & 14 \end{array}\right].} \end{array}}
From here, we can tell that the corresponding system is inconsistent.
Hence, this vector is not in the range 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 T.}

Final Answer:
(a)     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 [T]=\begin{bmatrix} 5 & -2.5 &10 \\ -1 & 0.5 & -2 \end{bmatrix}}
(b)     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 \begin{bmatrix} 45 \\ -9 \end{bmatrix}}
(c)     See above

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