copy of Matrix transformations (WBQ 1.30 randomised)

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Question 1

Let $A=\var{A}$ , $B=\var{B}$ , $u=\var{u}$ and $v=\var{v}$ .

a)

Compute:

$Au =$

Rows: Columns:

Expected answer:

$Av=$

Rows: Columns:

Expected answer:

$Bu=$

Rows: Columns:

Expected answer:

$Bv=$

Rows: Columns:

Expected answer:

Score: 0/4

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b)

The matrix transformation of $A$ is $T_A\colon { \mathbb{R}^n \to \mathbb{R}^m}$ for which numbers $n$ and $m$ ?

$n =$ and $m=$ Expected answer:

Score: 0/2

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c)

For a general vector $x=\begin{pmatrix}x_1\\x_2\\\vdots \\x_n\end{pmatrix}$ , determine $T_A(x)$ . Input $x_1$ as x_1 or x1, etc. Use the same notation for all variables, don't mix it.

$T_A(x)= \left(\begin{matrix} \phantom{.}\\\phantom{.}\\\phantom{.}\end{matrix} \right.$	interpreted as $\displaystyle{}$ Expected answer:interpreted as $\displaystyle{3 x_{1} + 2 x_{2} + 3 x_{3}}$	$\left.\begin{matrix} \phantom{.}\\\phantom{.}\\\phantom{.}\end{matrix} \right)$
	interpreted as $\displaystyle{}$ Expected answer:interpreted as $\displaystyle{-3 x_{2} - x_{3}}$

Score: 0/3

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Advice

a) To calculate matrix times vector $Ax$ , we calculate this row by row: the first entry of the resulting vector is "row 1 of the matrix times the vector". In general:

$\begin{pmatrix} a_{11} & a_{12} &a_{13} \\ a_{21} & a_{22} & a_{23}\end{pmatrix} \begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix}= \begin{pmatrix} a_{11}x_1+a_{12}x_2 +a_{13}x_3\\ a_{21}x_1+a_{22}x_2 +a_{23}x_3\end{pmatrix}$

So the vector $x$ needs to have as many entries as the matrix has columns, and the resulting vector $Ax$ has as many entries as $A$ has rows.

$\var{A}\var{u}=\var{unresolvedAu}=\var{A*u}$

$\var{A}\var{v}=\var{unresolvedAv}=\var{A*v}$

$\var{B}\var{u}=\var{unresolvedBu}=\var{B*u}$

$\var{B}\var{v}=\var{unresolvedBv}=\var{B*v}$

b) As explained above, the matrix $A$ can take a vector $x$ with $\var{n}$ entries as input, and gives a vector $Ax$ with $\var{n}$ entries as output. This is a matrix transformation $T_A\colon { \mathbb{R}^{\var{n}} \to \mathbb{R}^{\var{m}}}$ , i.e. $n=\var{n}$ and $m=\var{m}$ .

c) As $n=\var{n}$ , we calculate $\var{A}\var{latexx} = \var{latexAx}$

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copy of Matrix transformations (WBQ 1.30 randomised)

a)

b)

c)

Advice