Independent/Dependent Matrix Multiplication Proofs

[tpiaswordfish]

Independent Matrix Multiplication Proofs

Hello! This is my first time posting on the forum, and I hope I've picked the right section. I couldn't find a section specifically for linear algebra, but I saw a few other questions about matrices and vectors here, so this is where I decided to post. Please let me know if there's somewhere else I should be posting this!

Right now, I need help on a couple of proofs for my Elementary Computational Linear Algebra class. The two questions, which are almost identical, are worded like this:

• Prove or disprove: If the columns of B in R^(n*p) are linearly independent then so are the columns of AB (for A in R^(m*n).
• Prove or disprove: If the columns of B in R^(n*P) are linearly independent as well as those of A, then so are the columns of AB (for A in R^(m*n).

The notation confused me a bit, since my professor doesn't seem to use the same notation as the textbook uses. I think that, for example, "B in R^(n*p)" is saying that B is an n by p matrix comprised of real numbers, but correct me if I'm wrong! I have a gut feeling that the first can be disproven and the second can be proven, but I am just bad at quite figuring out why in mathematical language, no matter how much I play with the definitions.

Any help you can give me would be greatly appreciated. Thanks!

Edit: Found a typo after posting.

 

[HallsofIvy]

Hello! This is my first time posting on the forum, and I hope I've picked the right section. I couldn't find a section specifically for linear algebra, but I saw a few other questions about matrices and vectors here, so this is where I decided to post. Please let me know if there's somewhere else I should be posting this!

Right now, I need help on a couple of proofs for my Elementary Computational Linear Algebra class. The two questions, which are almost identical, are worded like this:

• Prove or disprove: If the columns of B in R^(n*p) are linearly independent then so are the columns of AB (for A in R^(m*n).

With no conditions on A, this is clearly not true. Take A to be the 0 matrix.

• Prove or disprove: If the columns of B in R^(n*P) are linearly independent as well as those of A, then so are the columns of AB (for A in R^(m*n).

The columns of a matrix are independent if and only if its determinant is non-zero.

The notation confused me a bit, since my professor doesn't seem to use the same notation as the textbook uses. I think that, for example, "B in R^(n*p)" is saying that B is an n by p matrix comprised of real numbers, but correct me if I'm wrong! I have a gut feeling that the first can be disproven and the second can be proven, but I am just bad at quite figuring out why in mathematical language, no matter how much I play with the definitions.

Any help you can give me would be greatly appreciated. Thanks!

Edit: Found a typo after posting.