Geometry of Linear Systems: True or False?

[FrogginTeach]

I am taking a Linear Algebra class to finish up my master's degree in math curriculum and instruction. I have been doing okay until these questions. I need some help.

True/False

1. If Ax=b has infinitely many solutions, then so does Ax=0.

2. If Ax=b is inconsistent, then Ax=0 has only the trivial solution.

3. The fewest number of hyperplanes in R^4 that can intersect in a single pt is 4.

4. If W is any plane in R^3, there is a linear system in three unknowns whose solution set is that plane.

5. If Ax=b is consistent, then every vector in the solution set is orthogonal to every row vector of A.

I really hope that someone can help me!

 

[Defennder]

I am taking a Linear Algebra class to finish up my master's degree in math curriculum and instruction. I have been doing okay until these questions. I need some help.

I'm a little surprised you haven't done a Linear Algebra class despite the fact you have a bacherlor's degree in math education.

1. If Ax=b has infinitely many solutions, then so does Ax=0.

If Ax=b has infinitely many solutions what does that imply about the reduced-row echelon form (RREF) of A? And what does that, in turn, tell you about how many solutions are there to Ax=0?

2. If Ax=b is inconsistent, then Ax=0 has only the trivial solution.

Like above, what does Ax=b being inconsistent imply about the RREF of A? And what does that tell you about Ax=0?

3. The fewest number of hyperplanes in R^4 that can intersect in a single pt is 4.

What is the general form of an equation of a hyperplane in R4? Suppose you have 3 or less hyperplanes represented as hyperplane equations. What does the intersection point of the hyperplanes tell you about solution set of the hyperplane equations?

4. If W is any plane in R^3, there is a linear system in three unknowns whose solution set is that plane.

How is a plane described in R3? How else may it be given?

5. If Ax=b is consistent, then every vector in the solution set is orthogonal to every row vector of A.

What does the definition of two vectors being orthogonal imply for any given vector in the solution set of the matrix equation Ax=b? More specifically, what happens if you have a vector v which is orthogonal to all the row vectors of A? Does it still satisfy Av=b?

 

[FrogginTeach]

I have taken Linear Algebra, but it has been a long time. I'm taking this course online and it is a lot harder to understand.

 

[FrogginTeach]

I'm still a little confused.

 

[Defennder]

Which one are you confused with? You need to be more specific.

 

[FrogginTeach]

Ok. Here are my answers

1. True
2. True
3. False
4. False
5. True

I'm not 100% confident in any of these answers.

 

[Defennder]

Any reason why you chose them as true/false for each one? I'm sure the question requires either a proof or a counterexample for each claim.

 

[FrogginTeach]

These homework questions are simply true or false questions requiring no work or explanation.

 

[FrogginTeach]

Okay. I decided to just choose systems to look at for a few of these problems.

True 1. I found that Ax = 0 would also have infinitely many solutions.

False 2. I found that Ax=0 would also be inconsistent.

I'm not even sure where to begin with the others.

 

[Defennder]

My answers for the first two are the same as yours. But not for the last 3. You only need a counterexample to disprove the false claims.