Linear Algebra - Spans and Linear Independence

[sassie]

Homework Statement

Let A and B be vector spaces, T:A->B be a linear transformation.

Give examples of:

(a) T, where a(1),... a(n) are linearly independent vectors in A, but T(a(1)),...T(a(n)) are not.
(b) T, where T(a(1)),...T(a(n)) span the range of T, but a(1),... a(n) do not span A.

Homework Equations

My ideas were to think about onto and 1-to-1-ness. (e.g. T is 1-to-1 iff the columns of the standard associated matrix T are linearly independent). However, I'm not 100% sure because the equations I have don't really make sure, and I'm not sure whether they apply to vector spaces.

The Attempt at a Solution

Let T be:

(a) [1 2 4 0
2 0 2 0
3 1 1 0]

(b) [1 2 3
0 2 5
0 0 2
0 0 0]

 

[slider142]

Those are good matrices, but don't forget the a(i)'s.

 

[sassie]

So my thinking is correct?

Sorry, but what do you mean by "a(i)'s"?

Cheers.

 

[slider142]

So my thinking is correct?

Sorry, but what do you mean by "a(i)'s"?

Cheers.

Yep. It asks for a(1),...,a(n) that satisfy those requirements.

 

[sassie]

Okay, so it my answer to (a) correct because it contains the zero vector? If not, how do I go about doing it?
This was the part of the question I was most confused about. I get the onto and 1-to-1 part for T not being linear independence and T spanning the range of T, but I don't get the a(1)...a(n) part. Can you help me on that part?

 

[slider142]

Okay, so it my answer to (a) correct because it contains the zero vector? If not, how do I go about doing it?
This was the part of the question I was most confused about. I get the onto and 1-to-1 part for T not being linear independence and T spanning the range of T, but I don't get the a(1)...a(n) part. Can you help me on that part?

Your answer to (a) is correct because it is a linear map from a 4-dimensional space to a 3-dimensional space. It is trivial that there do not exist 4 linearly independent vectors in a 3-dimensional space. You thus need only show 4 linearly independent vectors in the domain, knowing that they cannot possibly be mapped to 4 linearly independent vectors in the range.

 

[sassie]

And how would I do that? Do I need to take vectors out of the given matrix, or do I need to make up new examples?