Question

Suppose that T : R3 → R2 is given by:

Answer 1

Answer:  The required answers are

(a) T is proved to be a linear transformation.

(b) The matrix A such that T(x) = Ax is $\begin{pmatrix}1 & 0 &0 \\ 0 & 1 &0 \end{pmatrix}$

Step-by-step explanation:  We are given a linear transformation T : R³ → R² defined as follows :

$T(a,b,c)=(a,b).$

We are to

(a) prove that T is a linear transformation

and

(b) find a matrix A such that T(x) = Ax.

(a) Let s, t are any real numbers and (a, b, c), (a', b', c') ∈ R³.

Then, we have

$T(s(a,b,c)+t(a',b',c'))\\\\=T(sa+ta',sb+tb',sc+tc')\\\\=(sa+ta',sb+tb')\\\\=(sa,sb)+(ta'+tb')\\\\=s(a,b)+t(a',b')\\\\=sT(a,b,c)+tT(a',b',c').$

So, we get

$T(s(a,b,c)+t(a',b',c'))=sT(a,b,c)+tT(a',b',c').$

Therefore, T is a linear transformation.

(b) We know that B = {(1, 0, 0), (0, 1, 0), (0, 0, 1)} is a standard basis for R³ and B' = {(1, 0), (0, 1)} is a standard basis for R².

So, we have

$T(1,0,0)=(1,0)=1(1,0)+0(0,1),\\\\T(0,1,0)=(0,1)=0(1,0)+1(0,1),\\\\T(0,0,1)=(0,0)=0(1,0)+0(0,1).$

So, the matrix A such that T(x) = Ax will be given by

$\begin{pmatrix}1 & 0 &0 \\ 0 & 1 &0 \end{pmatrix}$

Thus,

(a) T is proved to be a linear transformation.

(b) The matrix A such that T(x) = Ax is  $\begin{pmatrix}1 & 0 &0 \\ 0 & 1 &0 \end{pmatrix}$