Answer to Question #292371 in Linear Algebra for praneeth

Question

Let T

 be a function from R

3

→R

3

 defined by

T(x,y,z)=(x−y+2z,2x+y,−x−2y+2z)

(i)

(ii)

  Show that T is a Linear Transformation

  Find nullity of T

Answer 1

A transformation T is linear if

1) T(ax)= aT(x)

2) T(x)=A(x) where A is a matrix.

Part 1

Ta(x,y,z)$=$ T(ax,ay,az)

$=(ax-ay+2az,2ax+ay,-ax-2ay+2az)$

$=a(x-y+2z,2x+y,-x-2y+2z)$

$=aT(x,y,z)$

$T(x,y,z)=x\begin{pmatrix} 1 \\ 2\\ -1 \end{pmatrix}+y\begin{pmatrix} -1 \\ 1 \\ -2 \end{pmatrix}+z\begin{pmatrix} 2 \\ 0 \\ 2 \end{pmatrix}$

$T(x,y,z)=\begin{pmatrix} 1&-1&2 \\ 2&1&0 \\ -1&-2&2 \end{pmatrix}$ $\begin{pmatrix} x \\ y\\ z \end{pmatrix}$

Therefore T is linear

Part 2

$\frac{1}{3}(R_2-2R_1\to\>R_2)$

$R_3+R_1\to\>R_3$

$\begin{pmatrix} 1&-1& 2\\ 1&1 & \frac{-4}{3}\\ 0&-3&4 \end{pmatrix}$

$R_1+R_2\to\>R_1$

$R_3+3R_2\to\>R_3$

$\begin{pmatrix} 1&0&\frac{2}{3} \\ 0&1&\frac{-4}{3} \\ 0&0&0 \end{pmatrix}$

Nullity is $=1$