Linear Algebra Question, Vector Images

[hemsley]

Homework Statement

Let T:R^4->R^3 be the linear transformation defined by

T( x1, x2,x3,x4) =
2(x1) - 4(x3)
(x2) -(x3)+3(x4)
(x1)+(x2)-3(x3)+2(x4)

Find the vector from the domain, Xd, which gives the image Xr = (2 1 1) in the range of T

The Attempt at a Solution

I don't need to necessarily know the values, I just need to know the process to get the 3 space matrix to find a vector in 4 space. To go from the Vector in Xd to the image Xr, I would just need to use the numbers in the matrix transform, or just multiply the standard matrix with the vector, but I am not sure how to go the other way. Do I take the inverse of the standard matrix multiply it by Xr possibly? Please help!

 

[micromass]

So you need to find $(x_1,x_2,x_3,x_4)$ such that

$$T(x_1,x_2,x_3,x_4)=(2,1,1)$$

Or

$$(2x_1-4x_3,x_2-x_3+3x_4,x_1+x_2-2x_3+2x_4)=(2,1,1)$$

Can you make a system of equations out of this??

 

[ehild]

You need to use the the relation between the 4D vector (x1,x2,x3,x4) and the image vector (2,1,1):

2(x1) - 4(x3)=2
(x2) -(x3)+3(x4)=1
(x1)+(x2)-3(x3)+2(x4)=1.

This is a system of linear equations, solve it with some standard method. The matrix of the linear transformation is

T=
2 0 -4 0
0 1 1 3
1 1 -3 2

It can not be inverted.ehild

Edit: Micromass beat me ...

 

[hemsley]

Thanks very much to both of you!