Do you understand that there are an infinite number of possible solutions? You are only asked to find one of them.

First, find a basis for W. W is the set of all such that and From those, we have that and . That is, so W is a two dimensional subspace of and {(1, 0, 3, -1), (0, 1, 1, 1)} is a basis for W.

We are told that U is spanned by (1, 0, 3, -1), (1, -1, 1, 1), and (1, 1, 5, -3). If such a linear tranformation exists, thosecannotbe independent. This linear transformation is on so the dimensions of it kernel and image must add to 4. Since W is two dimensional, U must also be 2 dimensional.

Of course, we check for independence by looking at a(1, 0, 3, -1)+ b(1, -1, 1, 1)+ c(1, 1, 5, -3)= (0, 0, 0, 0). That gives four equations: a+ b+ c= 0, -b+ c= 0, 3a+ b+ 5c= 0, and -a+ b- 3c= 0. Adding the first two equations, a+ 2c= 0. Adding the second and third equations also eliminates b: 3a+ 6c= 0. Dividing that last equation by 3 gives a+ 2c= 0 so those are really the same equation. Putting a= -2c into a+ b+ c= 0 gives -2c+ b+ c= b- c= 0 or b= c. That is, for any value of c, a= -2c and b= c satisfy that equation. In particular, taking c= 1, -2(1, 0, 3, -1)+ (1, -1, 1, 1)+ (1, 1, 5, -3)= (0, 0, 0, 0). We can take any two of them, say {(1, 0, 3, 1), (1, -1, 1, 1)} as basis for U.

Such a linear transformation must be from to and so can be written as a 4 by 4 matrix. The we know a basis for the kernel, U. Find two more vectors in that are independent of those two as well as independent of each other. In fact, it is easy to see that (1, 0, 0, 0) and (0, 1, 0, 0) are independent of (1, 0, 3, -1) and (0, 1, 1, 1) so {(1, 0, 0, 0), (0, 1, 0, 0), (1, 0, 3, -1), (0, 1, 1, 1) is a basis for

Write A as .

Now we want

and

That gives 16 equations to solve for the 16 entries of A, but they simplify a lot. For example, the first equation you get is just a= 1 and the second equation is e= 0.

2.

L={(,...)|=+}

Find basis which elements are geometrical progression.

3.This the matrix A:

3 -1 1

4 -1 2

2 -1 2

Find such T that the matrix D = is diagonal.Find D.