Linear Algebra Problems (Easy?)

In summary, the conversation discusses a homework problem with two parts. The first part asks to prove that the equation Ax=b has a unique solution if and only if the solution to Ax=0 is x=0. The second part involves two matrices, A of size m x p and B of size p x n, and asks to show that the range of A is contained in the range of AB and the kernel of B is contained in the kernel of AB. The conversation includes attempts at solving the first problem and discussing the strategy for the second problem.
  • #1
gasaway.ryan
7
0
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.

Homework Statement



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?

Homework Equations



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

2. See below.

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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  • #2
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?
 
  • #3
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.
 
  • #4
You've got the right idea. The phrasing could use some work. You assumed x1 and x2 are two different solutions, you conclude A(x1-x2)=0.
Ok but now don't "assume x1=x2". You've got x1-x2=/=0. That's another solution to Ax=0 besides x=0. So you've got "if the solution to Ax=b isn't unique then the solution to Ax=0 isn't unique". That's the same as saying "the solution to Ax=0 is unique implies the solution to Ax=b is unique", right? Now since the question says "if and only if" you should do the opposite direction as well.

For the second one, think of the matrices as linear maps. A maps R^p to R^m. AB maps R^n to R^m by going through R^p. Look up the definition of 'range' and think about it.
 

1. What is linear algebra?

Linear algebra is a branch of mathematics that deals with the study of linear equations and their representations in vector spaces. It involves solving systems of linear equations, performing operations on matrices, and analyzing geometric transformations.

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Linear algebra has a wide range of applications in various fields such as engineering, physics, computer science, and economics. Some common examples include image and signal processing, machine learning, cryptography, and economic modeling.

3. What are the basic operations in linear algebra?

The basic operations in linear algebra include addition, subtraction, multiplication, and division of vectors and matrices. Other important concepts include determinants, eigenvalues and eigenvectors, and matrix factorization methods like LU decomposition and QR decomposition.

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5. Are there any resources available for practicing linear algebra problems?

Yes, there are many resources available for practicing linear algebra problems. These include textbooks, online courses, video tutorials, and practice worksheets. Additionally, many universities and colleges offer free resources for practicing linear algebra problems, such as past exams and solutions.

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