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:

Step 2:

(b)

Step 1:

Step 2:

(c)

Step 1:

Step 2:

Final Answer:
(a)
(b)

Return to Sample Exam