## Linear Algebra Problems (Easy?)

This stuff is confusing. I don't know if it's hard or not, I just have a feeling I don't really know what I'm doing.

1. The problem statement, all variables and given/known data

1. Show that the equation Ax=b has a unique solution if and only if the solution to Ax=0 is x=0.

2. Let A be an m x p matrix, and let B be a p x n matrix. Show that the range of A is contained in the range of AB. Show that the kernel of B is contained in the kernel of AB. Is the reverse inclusion true in either case?

2. Relevant equations

1. Ax=b; Ax=0; x=0

2. See below.

3. The attempt at a solution

1. I really have no idea what to do.

2. A=[v1 ... vp]; B=[w1 ... wn]

AB=[Aw1 ... Awn]

im(A)=c1v1 + ... + cpvp

im(AB)=c1Aw1 + ... + cnAwn=A(c1w1 + ... + cnwn)

That's all I have. This is probably not even close to what I'm supposed to be doing. Please help!
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 Recognitions: Homework Help Science Advisor No, you don't know what you are doing. You need practice. Let's just start with the first one. Suppose Ax=b has two different solutions. Ax1=b and Ax2=b with x1 not equal to x2. If you subtract those two equations what does that tell you about solutions to Ax=0?
 Okay, here's my attempt at solving the first. Suppose Ax=b has two solutions, x=x1 and x=x2, where x1=/=x2. Ax1=b Ax2=b Ax1=Ax2 -Ax2 -Ax2 Ax1-Ax2=0 A(x1-x2)=0 Now let's assume x1=x2; this condition implies that Ax=b has a unique solution. A(x1-x2)=0 A0=0 We thus see that the equation Ax=b has a unique solution if and only if the unique solution to Ax=0 is x=0. Is this extraneous or wrong? Also, what are you thoughts on the second problem? Any suggestions for practice? Thanks so much.

Recognitions:
Homework Help