Linear Algebra - Column and Null Space

[vwishndaetr]

Homework Statement

Ax=b where,

A = 2 -1
...-1 2

Homework Equations

a) Find Null Space N(A) and Column Space C(A)
b) For which vectors b does the system Kx=b have a solution?
c) How many solution x does the system have for any given b?

The Attempt at a Solution

a)

For Null Space, I set Ax=0, and got the zero vector. No problem with that.
For Column Space, I set Ax=b, and got b = span ( 2 , -1 )
............-1 2
No problem with Column Space either.
Those are two vectors, I am rusty with notation on here.

b) This is simply the Column Space. All vectors b that span R² allow Kx=b to have a solution. Ni problem with that either.

c) This is where my issue is. For a given b, how many solutions does x have? No solutions? One? Infinitely many?

I am having a hard time sorting this out intuitively. I want to say it will be one solution for a given b, but I can't come to that agreement.

 

[Dick]

Suppose Kx=b has two solutions, Ka1=b and Ka2=b. Subtract the two equations. What can you say about a1-a2 vis a vis a space related to K that it might be in?

 

[vwishndaetr]

Bit confused about what you mean by vis.

 

[HallsofIvy]

"vis a vis" means "in relation to".

His point was that if Ax= b and Ay= b then Ax- Ay= b- b or A(x-y)= 0. That is, x- y is a vector in the kernel of A. What did you get for the kernel of A?

 

[vwishndaetr]

Kernel is related to Null Space correct?

So, since I got the zero vector for Null Space, is the only solution the trivial one?

A(x-y) = 0, where x-y = 0 = N(A) ?

 

[Dick]

Kernel is related to Null Space correct?

So, since I got the zero vector for Null Space, is the only solution the trivial one?

A(x-y) = 0, where x-y = 0 = N(A) ?

Yes. x-y=0 means x=y.

 

[HallsofIvy]

Kernel is related to Null Space correct?

So, since I got the zero vector for Null Space, is the only solution the trivial one?

A(x-y) = 0, where x-y = 0 = N(A) ?

"Kernel" is another word for "null space". Sorry, I should have noticed that you used "null space".

Yes, you are correct that the null space for this operator is {0}. So in order that Ax=b= Ay, which is the same as A(x-y)= 0, x- y must be in the null space: x-y= 0 or x= y. So how many different solutions can Ax= b have?

 

[vwishndaetr]

Please forgive me, but still a wee bit confused.I want to say one solution where x = y = 0.

They way I am thinking is,

Let Az = 0, then z = zero vector.

So, A(x-y) = A(z) = 0

then, x-y = z = 0, so x = y

So infinitely many solutions?

I really want to understand this, because I am a few problems similar so I do not want to just "accept" it. But it feels like I am walking in circles. Maybe it is easier than I what I expect?

 

[Dick]

Please forgive me, but still a wee bit confused.I want to say one solution where x = y = 0.

They way I am thinking is,

Let Az = 0, then z = zero vector.

So, A(x-y) = A(z) = 0

then, x-y = z = 0, so x = y

So infinitely many solutions?

I really want to understand this, because I am a few problems similar so I do not want to just "accept" it. But it feels like I am walking in circles. Maybe it is easier than I what I expect?

The point is that if A(x)=b and A(y)=b, then A(x-y)=0 (since Ax-Ay=b-b), ok so far? If A(x-y)=0 then x-y is in the null space, still ok? Since you've shown the null space is {0}, then x-y=0. So x=y. So if x and y are both solutions, then x-y. Now, how many solutions to Ax=b?

 

[vwishndaetr]

The point is that if A(x)=b and A(y)=b, then A(x-y)=0 (since Ax-Ay=b-b), ok so far? If A(x-y)=0 then x-y is in the null space, still ok? Since you've shown the null space is {0}, then x-y=0. So x=y. So if x and y are both solutions, then x-y. Now, how many solutions to Ax=b?

ohhhh. Ohhhhh.

Duh.

3 solutions. x is a solution, y is a solution, and x-y = 0 is a solution.

Do I have it now?

 

[Dick]

i) 0 isn't a solution to Ax=b (unless b=0, of course). ii) Sure, x is a solution and y is a solution. But x=y! You've shown any two solutions are EQUAL. Take a walk around the block and think this over for a bit. Then count them again.

 

[HallsofIvy]

ohhhh. Ohhhhh.

Duh.

3 solutions. x is a solution, y is a solution, and x-y = 0 is a solution.

Do I have it now?

NO! For one thing, x- y= 0 is not a solution to Ax= b, which is what you are working with.
And having said that x- y= 0, it follows that x=y. In other words, x and y are NOT different solutions.

 

[vwishndaetr]

So that means there is a unique solution for every b.

 

[Dick]

Yes. Any problems with that?

 

[vwishndaetr]

Yes. Any problems with that?

No that is it for me; crystal clear now. Sorry for being so hard-headed.

I am going to do one more, and I will come back to make sure I understand everything correctly. Thanks for the help! This beats any other sources! :)