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

$V_1=[1,1,1]$

$V_2=[0,1,1]$

$V_3=[0,0,1]$

$T=[x-y+z,x+y,y+z]$

According to this the new

$V_1'=[1,2,2]$

$V_2'=[0,1,2]$

$V_3'=[1,0,1]$

So the matrix T form is as follows:

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

For the invertible matrix the determinant of the matrix is = 0

Det(T) = $1(1-0)-2(0-2)+2(0-1)=1+4-2=3$

As the determinant of the matrix is not equal to zero hence the given matrix is invertible.