Linear Algebra - Spans and Linear Independence

In summary: 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.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?
  • #1
sassie
35
0

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]
 
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  • #2
Those are good matrices, but don't forget the a(i)'s.
 
  • #3
So my thinking is correct?

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

Cheers.
 
  • #4
sassie said:
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.
 
  • #5
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?
 
  • #6
sassie said:
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.
 
  • #7
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?
 

1. What is the definition of a span in linear algebra?

A span in linear algebra is the set of all possible linear combinations of a given set of vectors. It is represented as a subspace of the vector space generated by those vectors.

2. How do you determine if a set of vectors is linearly independent?

A set of vectors is linearly independent if no vector in the set can be written as a linear combination of the other vectors. This means that the only solution to the equation c1v1 + c2v2 + ... + cnvn = 0, where c1, c2, ..., cn are scalars and v1, v2, ..., vn are the vectors in the set, is c1 = c2 = ... = cn = 0.

3. What is the difference between a linearly independent set and a spanning set?

A linearly independent set is a set of vectors that cannot be written as a linear combination of each other. A spanning set, on the other hand, is a set of vectors that can generate all possible vectors in a given vector space through linear combinations. A linearly independent set can be a spanning set, but a spanning set may not necessarily be linearly independent.

4. Can a set of vectors span a vector space without being linearly independent?

Yes, a set of vectors can span a vector space without being linearly independent. This means that the vectors in the set are not unique and can be written as linear combinations of each other. However, they can still generate all possible vectors in the vector space through these linear combinations.

5. How do you find the basis of a vector space?

To find the basis of a vector space, you need to first find a set of linearly independent vectors that span the vector space. This set of vectors will form the basis of the vector space. You can also find the basis by finding the pivot columns in the reduced row echelon form of the matrix representing the vector space.

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