Homework Help: Linear Algebra Question, Vector Images

1. Dec 13, 2011

hemsley

1. The problem statement, all variables and given/known data
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

3. The attempt at a solution
I dont 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 Im not sure how to go the other way. Do I take the inverse of the standard matrix multiply it by Xr possibly? Please help!

2. Dec 13, 2011

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??

3. Dec 13, 2011

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 ...

4. Dec 13, 2011

hemsley

Thanks very much to both of you!