Invertible, kernel, and range of a Linear Transformation

[hkus10]

1) Let L:R3 >>>R3 be defined by
L([1 0 0]) = [1 2 3],
L([0 1 0]) = [0 1 1],
L([0 0 1]) = [1 1 0]

How to prove that L is invertible? I have the idea of one-to-one and onto, but I do not know how to apply them to this proof.

2) Find a linear transformation L:R2 >>>R3 such that {[1 -1 2], [3 1 -1]} is a basis for range (L). How can I approach this problem?

3) Let S = {v1,…,vn} be an ordered basis for vector space V . Let
L :V →R^n be given by L(v) = [v]S . Prove that L is an
isomorphism( prove that the linear transformation is one-to-one and onto)
What I know so far is that {[v1]s, [v2]s,...,[vn]s} is an ordered basis for R^n and v can be written in a unique way sych that v = a1v1+...+anvn = 0.
How can I go from there?

4) If L:V>>>W is a linear transformation of a vector space V into a vector space W and dim V = dim W, then prove that if L is one-to-one, then it is onto.
I get a point that dim(W) = dim(range(L)). What I am trying to get to is W = range(L). Then, by definition, L is onto. My question is that how can I prove that dim(W) = dim(range(L)) >>> W = range(L) and by what Thm of defin? If not, how should I approach this question?

5) Let L:R^n>>>R^m be a linear transformation defined by L(x) = Ax, for x in R^n. Then, L is onto if and only if rank A = m.
In this question, is the nullity of A equal to the nullity of L?

Thanks

 

[upsidedowntop]

1)
What is L's matrix representation with respect to the standard basis? If you can show that matrix representation to be invertible, then L is also invertible.

2)
Look up the rank-nullity theorem.

3)
Choose some basis for R^2, then define a map from that basis to the desired basis for the image. The action of a linear map on a basis uniquely defines that linear map, so you're done.

4)
I'm not totally sure I follow the question. But I think you've been asked to show that if L is defined as mapping vectors v to the coordinates of their representation in a basis S, then L is an isomorphism.

From what you have, if \{ [v_1]_s, ..., [v_n]_s \} is a basis for R^n, what does that tell you about the image of L?

Good luck with your work!

 

[hkus10]

2)
Look up the rank-nullity theorem.

3)
Choose some basis for R^2, then define a map from that basis to the desired basis for the image. The action of a linear map on a basis uniquely defines that linear map, so you're done.

For 2) Is the answer just simply 2 + 5?

For 3) what do you mean by a linear map?

Also, I have an additional question which is If L:V>>>W is a linear transformation of a vector space V into a vector space W and dim V = dim W, then prove that if L is one-to-one, then it is onto.
I get a point that dim(W) = dim(range(L)). What I am trying to get to is W = range(L). Then, by definition, L is onto. W = range(L). My question is that how can I prove that dim(W) = dim(range(L)) >>> W = range(L) and by what Thm of defin? If not, how should I approach this question?

 

[upsidedowntop]

For 2) Is the answer just simply 2 + 5?

For 3) what do you mean by a linear map?

Also, I have an additional question which is If L:V>>>W is a linear transformation of a vector space V into a vector space W and dim V = dim W, then prove that if L is one-to-one, then it is onto.
I get a point that dim(W) = dim(range(L)). What I am trying to get to is W = range(L). Then, by definition, L is onto. W = range(L). My question is that how can I prove that dim(W) = dim(range(L)) >>> W = range(L) and by what Thm of defin? If not, how should I approach this question?

2) Yes. It's an easy question assuming rank-nullity.

3) Linear map and linear transformation are synonymous. If one runs into mathematical terminology one does not recognize, reading the first paragraph of the wikipedia page for the term will usually sort out one's difficulty. For example, linear map and linear transformation both send you to the same wikipedia page that begins with
""""
In mathematics, a linear map, linear mapping, linear transformation, or linear operator (in some contexts also called linear function) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. The expression "linear operator" is commonly used for linear maps from a vector space to itself (i.e., endomorphisms).
"""

The implication you suggest proving for the last question is true. It might be easier to prove the following more general statement though.
Let V be a finite dimensional vector space with vector subspace W. Then dim(W) = dim(V) implies W = V.

 

[hkus10]

push!