Question

Let

T:

R3→R2

be a linear transformation such that

(1,1,1) = (1,0),

(1,1,0) = (2, −1) and

(1,0,0) =

(4,3). What is

(2, −3,5)?

Answer 1

Find a1 , a2 and a3 such that $v = a_{1} v_{1} + a_{2} v_{ 2} + a_{3}v_{3},$

The above will lead to a linear system whose augmented matrix is:

$\left[\begin{array}{rrr|r} 1 & 1 & 1& 2\\ 1 & 1& 0& -3\\ 1 & 0& 0& 5 \end{array}\right]$

 Transform the above matrix to reduced row echelon form, and find the following values:

$a_{1} = 5\\ a_{2} = -8\\ a_{3} = 5$

Therefore,

$v = 5v_{1} – 8v_{2} + 5v_{3}$

$v = 5[1,0] – 8[2, -1] + 5[4,3]\\ v= [9, 23]$

Therefore, (2, −3,5) = (9,23)