# Math Help - Determining Linear Independence

1. ## Determining Linear Independence

[Solved, thanks]

Hi. My Linear Algebra with Applications, 4th Ed. by Otto Bretscher (Pearson International Edition) poses the following exercise:

Consider linearly independent vectors v1, v2, ..., vm in R(n), and let A be an invertible mxm matrix. Are the columns of the following matrix linearly independent?

[v1 | v2 | ... | vm]A

What I know relevant to linear independence is that the following statements are equivalent, for a list v1, v2, ..., vm of vectors in R(n):
1. Vectors v1, v2, ..., vm are linearly independent
2. None of the vectors v1, v2, ..., vm is redundant
3. ker[v1 | v2 | ... | vm] = {0}
4. rank[v1 | v2 | ... | vm] = m

Can anyone give me pointers in what to be looking at in order to answer this question?

Thanks!

2. Originally Posted by Oijl
Hi. My Linear Algebra with Applications, 4th Ed. by Otto Bretscher (Pearson International Edition) poses the following exercise:

Consider linearly independent vectors v1, v2, ..., vm in R(n), and let A be an invertible mxm matrix. Are the columns of the following matrix linearly independent?

[v1 | v2 | ... | vm]A

What I know relevant to linear independence is that the following statements are equivalent, for a list v1, v2, ..., vm of vectors in R(n):
1. Vectors v1, v2, ..., vm are linearly independent
2. None of the vectors v1, v2, ..., vm is redundant
3. ker[v1 | v2 | ... | vm] = {0}
4. rank[v1 | v2 | ... | vm] = m

Can anyone give me pointers in what to be looking at in order to answer this question?

Thanks!
Suppose that $\displaystyle v_1=\sum_{k=2}^{m}\alpha_k v_k$ for example. Then, a pretty quick check shows that $\displaystyle A(e_1)=A\left(\sum_{k=2}^{m}\alpha_k e_k\right)$ where $\{e_1,\cdots,e_m\}$ is the canonical basis for $\mathbb{R}^m$ but by injectivity this impiles that $\displaystyle e_1=\sum_{k=2}^{m}\alpha_k e_k$ which is a contradiction.

3. Originally Posted by Oijl
Hi. My Linear Algebra with Applications, 4th Ed. by Otto Bretscher (Pearson International Edition) poses the following exercise:

Consider linearly independent vectors v1, v2, ..., vm in R(n), and let A be an invertible mxm matrix. Are the columns of the following matrix linearly independent?

[v1 | v2 | ... | vm]A

What I know relevant to linear independence is that the following statements are equivalent, for a list v1, v2, ..., vm of vectors in R(n):
1. Vectors v1, v2, ..., vm are linearly independent
2. None of the vectors v1, v2, ..., vm is redundant
3. ker[v1 | v2 | ... | vm] = {0}
4. rank[v1 | v2 | ... | vm] = m

Can anyone give me pointers in what to be looking at in order to answer this question?

Thanks!
Is m < n, m > n, or m = n?

If that isn't giving, you can look at each case.

4. Thanks for trying to help, but I got it now.