Trying to understand(conceptually) orthogonality

[Arnoldjavs3]

Homework Statement

if Ax = b has a solution and A^Ty = 0 , is y^T(x) = 0 or y^T(y) =0

Homework Equations

The Attempt at a Solution

I simply do not think i understand the properties to answer this question.

From my understandinging, the transpose of A times y is = 0. This means that A transpose and y(all the members of these two subspaces) are perpendicular. Does this indicate that y is essentially the nullspace of A transpose? A transpose is the rowspace?

Where do I progress on frmo this?

I'd like to add... why is the transpose of these matrices even relevant? I don't understand. Are there properties of the transpose that I'm missing? What are these variables y and x representing? There's just so much here that's confusing me(I'm guessing I don't understand fundamentally)

I know that the row space is orthogonal to the nullspace(and colspace with the left nullspace) how do i use this knowledge to solve this problem?

 

[andrewkirk]

Do you mean Ax=b has a unique solution?
If it has a unique solution then A is invertible, and hence so is $A^T$, which allows us to choose between the two alternative answers.

But if we only know that Ax=b has at least one solution, then A may be singular, in which case the question is unanswerable.

 

[Arnoldjavs3]

Yes it has a unique solution.

What good does it knowing that A^T and ax=b is invertible?

 

[andrewkirk]

You are given the equation $A^Ty=0$. Using the fact that $A$ is invertible, and hence so is $A^T$, rearrange that equation so that y is alone on one side (ie make y the subject of the equation). Then simplify as much as possible.

 

[Arnoldjavs3]

Well if I were to isolate y from that equation you've stated, wouldn't it just be y = 0? the fact that A is invertible(meaning so is its transpose) confuses me. Why does that help me?

 

[andrewkirk]

They've asked you if $y^Tx = 0$ or $y^Ty =0$. The answer is yes if either or both is true.
If you can show that $y=0$, can you then prove that at least one of those two is true?

If so, the answer to the set question is Yes.

 

[Mark44]

Homework Statement

if Ax = b has a solution and A^Ty = 0 , is y^T(x) = 0 or y^T(y) =0

What do y^T(x) and y^T(y) mean?
It would be helpful to state that x and y are vectors in some vector space (such as $\mathbb{R}^n$?).

Since your question is about orthoganality, there's an inner product lurking about somewhere, so presumably y^T(x) could just as well be written as $\vec{y} \cdot \vec{x}$ and y^T(y) could be written as $\vec{y} \cdot \vec{y}$

Homework Equations

The Attempt at a Solution

I simply do not think i understand the properties to answer this question.

From my understandinging, the transpose of A times y is = 0. This means that A transpose and y(all the members of these two subspaces) are perpendicular.

That doesn't make any sense. Two vectors can be perpendicular (or othogonal), but a matrix and a vector aren't things that can be perpendicular to each other.

Does this indicate that y is essentially the nullspace of A transpose? A transpose is the rowspace?

If A^Ty = 0, then, yes, y is in the nullspace of A^T.

Where do I progress on frmo this?

I'd like to add... why is the transpose of these matrices even relevant? I don't understand. Are there properties of the transpose that I'm missing? What are these variables y and x representing? There's just so much here that's confusing me(I'm guessing I don't understand fundamentally)

I know that the row space is orthogonal to the nullspace(and colspace with the left nullspace) how do i use this knowledge to solve this problem?

 

[vela]

Homework Statement

if Ax = b has a solution and A^Ty = 0 , is y^T(x) = 0 or y^T(y) =0

Homework Equations

The Attempt at a Solution

I simply do not think i understand the properties to answer this question.

From my understandinging, the transpose of A times y is = 0. This means that A transpose and y(all the members of these two subspaces) are perpendicular. Does this indicate that y is essentially the nullspace of A transpose? A transpose is the rowspace?

Where do I progress on frmo this?

I'd like to add... why is the transpose of these matrices even relevant? I don't understand. Are there properties of the transpose that I'm missing? What are these variables y and x representing? There's just so much here that's confusing me(I'm guessing I don't understand fundamentally)

I know that the row space is orthogonal to the nullspace(and colspace with the left nullspace) how do i use this knowledge to solve this problem?

Could you post the problem exactly as given to you? It seems like you're leaving a lot of relevant details out.