031 Review Part 2, Problem 3

Let  $B={\begin{bmatrix}1&-2&3&4\\0&3&0&0\\0&5&1&2\\0&-1&3&6\end{bmatrix}}.$

(a) Is  $B$  invertible? Explain.

(b) Define a linear transformation  $T$  by the formula  $T({\vec {x}})=B{\vec {x}}.$  Is  $T$  onto? Explain.

Foundations:
1. A matrix  $A$  is invertible if and only if  ${\text{det }}A\neq 0.$
2. A linear transformation  $T$  given by  $T({\vec {x}})=A{\vec {x}}$  where  $A$  is a  $m\times n$  matrix
if and only if the columns of  $A$  span  $\mathbb {R} ^{m}.$

Solution:

(a)

Step 1:

Step 2:

(b)

Step 1:

Step 2:

Final Answer:
(a)
(b)

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