Question

Define: R^3→R^3 by

T(x,y,z)=(x-y+z,x+y,y+z)

Let v1= (1,1,1), v2= (0,1,1), v3= (0,0,1). Find a matrix of T with respect to the basis {v1,v2,v3}. Futher check T is invertible or not.

Answer 1

Here T($x,y,z)=(x-y+z,x+y,y+z)$

$v_1=(1,1,1)$ , $v_2=(0,1,1)$ , $v_3=(0,0,1)$

So $T(1,1,1)=(1-1+1,1+1,1+1)=(1,2,2)$

$T(0,1,1)=(0-1+1,0+1,1+1)=(0,1,2)$

$T(0,0,1)=(0-0+1,0+0,0+1)=(1,0,1)$

So Matrix of T with respect to basis {$v_1,v_2,v_3$} is :

$A=\begin{bmatrix} 1& 2 &2 \\ 0 & 1 & 2\\ 1 &0&1 \end{bmatrix}$

Here |A|=$1(1-0)-2(0-2)+2(0-1)=1+4-2=5-2=3$ which is not equal to .So matrix is invertible.