find the assosiated matrix for the linear map Tr3 to r4 such

find the assosiated matrix for the linear map T:r^3 to r^4 such that T(x,y,x)=(x-y+3z,y+z,3x-2y-z,7y+z)

Solution

Let the Matrix A be the associated matrix which will be of the order 4x3 as the mapping is from R^3 to R^4 so the order of the associated matrix becomes 4x3.

we will write A as:

We need to satisfy f(x)=Ax

So we get:

T(X)=AX ,Where X=(x,y,z)

Now to find A:

Case1: Let X=(1,0,0) ,then T(X)=AX is the first column of A.

T(1,0,0)=(1-0+3(0),0+0,3(1)-2(0)-0,7(0)+0)=(1,0,3,0)

Case2: Let X=(0,1,0) ,Then T(X)=AX is the second column of A.

T(0,1,0)=(0-1+3(0),1+0,3(0)-2(1)-0,7(1)+0)=(-1,1,-2,7)

Case3:Let X=(0,0,1) ,Then T(X)=AX is the third column of A.

T(0,0,1)=(0-0+3(1),0+1,3(0)-2(0)-1,7(0)+1)=(3,1,-1,1)

Putting all these together, we see that the linear Transformation T(X) is associated with the Matrix

A=