031 Review Part 2, Problem 7

(a) 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 T:\mathbb{R}^2\rightarrow \mathbb{R}^2}   be a transformation given by

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\bigg( \begin{bmatrix} x \\ y \end{bmatrix} \bigg)= \begin{bmatrix} 1-xy \\ x+y \end{bmatrix}.}

Determine whether  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}   is a linear transformation. Explain.

(b) 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 A= \begin{bmatrix} 1 & -3 & 0 \\ -4 & 1 &1 \end{bmatrix}}   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= \begin{bmatrix} 2 & 1\\ 1 & 0 \\ -1 & 1 \end{bmatrix}.}   Find  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 AB,~BA^T}   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 A-B^T.}

Foundations:
A map  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:\mathbb{R}^n\rightarrow \mathbb{R}^m}   is a linear transformation 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 T(\vec{x}+\vec{y})=T(\vec{x})+T(\vec{y})}
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 T(a\vec{x})=aT(\vec{x})}
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 \vec{x},\vec{y}\in \mathbb{R}^n}   and 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 a\in \mathbb{R}.}

Solution:

(a)

Step 1:
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 T}   is not a linear transformation.
Consider the 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 \begin{bmatrix} 1\\ 0 \end{bmatrix}}   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 \begin{bmatrix} 0\\ 1 \end{bmatrix}.}
Then, 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\bigg(\begin{bmatrix} 1\\ 0 \end{bmatrix}+\begin{bmatrix} 0\\ 1 \end{bmatrix}\bigg)} & = & \displaystyle{T\bigg(\begin{bmatrix} 1\\ 1 \end{bmatrix}\bigg)}\\ &&\\ & = & \displaystyle{\begin{bmatrix} 0\\ 2 \end{bmatrix}.} \end{array}}

Step 2:
On the other hand, notice
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\bigg(\begin{bmatrix} 1\\ 0 \end{bmatrix}\bigg)+T\bigg(\begin{bmatrix} 0\\ 1 \end{bmatrix}\bigg)} & = & \displaystyle{\begin{bmatrix} 1\\ 1 \end{bmatrix}+\begin{bmatrix} 1\\ 1 \end{bmatrix}}\\ &&\\ & = & \displaystyle{\begin{bmatrix} 2\\ 2 \end{bmatrix}.} \end{array}}
So, now we know
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\bigg(\begin{bmatrix} 1\\ 0 \end{bmatrix}+\begin{bmatrix} 0\\ 1 \end{bmatrix}\bigg)\neq T\bigg(\begin{bmatrix} 1\\ 0 \end{bmatrix}\bigg)+T\bigg(\begin{bmatrix} 0\\ 1 \end{bmatrix}\bigg).}
Therefore,  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}   is not a linear transformation.

(b)

Step 1:
Using the row-column rule for multiplication, 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{AB} & = & \displaystyle{\begin{bmatrix} 1 & -3 & 0 \\ -4 & 1 &1 \end{bmatrix}\begin{bmatrix} 2 & 1\\ 1 & 0 \\ -1 & 1 \end{bmatrix}}\\ &&\\ & = & \displaystyle{\begin{bmatrix} 1(2)+-3(1)+0(-1) & 1(1)+-3(0)+0(1)\\ -4(2)+1(1)+1(-1) & -4(1)+1(0)+1(1) \end{bmatrix}}\\ &&\\ & = & \displaystyle{\begin{bmatrix} -1 & 1\\ -8 & -3 \end{bmatrix}.} \end{array}}

Step 2:
Now,  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}   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 A^T}   are both  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 3\times 2}   matrices.
Hence,  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 BA^T}   is undefined.

Step 3:
For  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^T,}   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{A-B^T} & = & \displaystyle{\begin{bmatrix} 1 & -3 & 0 \\ -4 & 1 &1 \end{bmatrix}-\begin{bmatrix} 2 & 1 & -1\\ 1 & 0 & 1\\ \end{bmatrix}}\\ &&\\ & = & \displaystyle{\begin{bmatrix} -1 & -4 & 1\\ -5 & 1 & 0 \end{bmatrix}.} \end{array}}

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}   is not a linear transformation
(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 AB=\begin{bmatrix} -1 & 1\\ -8 & -3 \end{bmatrix},}   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 BA^T}   is undefined 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 A-B^T=\begin{bmatrix} -1 & -4 & 1\\ -5 & 1 & 0 \end{bmatrix}}

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