What is the formula for the y-axis projection along a given line?

[jclawson709]

Homework Statement

Let T: R^2 -> R^2.
Part a: Find a formula for T(a,b) where T represents the projection on the y-axis along the x-axis.
Part b: Find a formula for T(a,b) where T represents the projection on the y-axis along the line L={(s,s):s is an element of R}

Homework Equations

My textbook says that T:V -> V is a projection on W1 along W2 (for W1 and W2 subspaces of V and V is the direct sum of W1 and W2) if, for x = x1 + x2 with x1 an element of W1 and x2 an element of W2, T(x) = x1

The Attempt at a Solution

The book gives a pretty lousy definition of projection in my opinion and we haven't talked about it in class (I think my professor is leaving it up to the book), so my attempt at a solution is pretty half-baked. I'm unsure of what the textbook means by "find a formula" but for part a I think T(a,b) = (a,0) and for part b i really have no idea. All I can find on the internet are matrix definitions which really don't help too much. Oh and the textbook I am using is Linear Algebra 4th edition by Friedberg, Insel, & Spence

 

[jclawson709]

oops i actually meant T(a,b) = (0,b) for part a

 

[LCKurtz]

Homework Statement

Let T: R^2 -> R^2.
Part a: Find a formula for T(a,b) where T represents the projection on the y-axis along the x-axis.
Part b: Find a formula for T(a,b) where T represents the projection on the y-axis along the line L={(s,s):s is an element of R}

Homework Equations

My textbook says that T:V -> V is a projection on W1 along W2 (for W1 and W2 subspaces of V and V is the direct sum of W1 and W2) if, for x = x1 + x2 with x1 an element of W1 and x2 an element of W2, T(x) = x1

The Attempt at a Solution

The book gives a pretty lousy definition of projection in my opinion and we haven't talked about it in class (I think my professor is leaving it up to the book), so my attempt at a solution is pretty half-baked. I'm unsure of what the textbook means by "find a formula" but for part a I think T(a,b) = (a,0) and for part b i really have no idea. All I can find on the internet are matrix definitions which really don't help too much. Oh and the textbook I am using is Linear Algebra 4th edition by Friedberg, Insel, & Spence

oops i actually meant T(a,b) = (0,b) for part a

So for part (a) presumably you wrote (a,b) = (0,b) + (a,0) = W₁ + W₂, so T(a,b) = W₁ = (0,b).

That is the way you worked it, right?

So you work part b the same way. Start by writing (a,b) as the sum of points in your two subspaces.

 

[jclawson709]

So for part (a) presumably you wrote (a,b) = (0,b) + (a,0) = W₁ + W₂, so T(a,b) = W₁ = (0,b).

That is the way you worked it, right?

So you work part b the same way. Start by writing (a,b) as the sum of points in your two subspaces.

Hmm no I just presumed intuitively that that was the answer, i didn't think it all the way out like that. So for part b, would I write W1 = (0,s), W2 = (s,0) so W1 + W2 = (s,s) and therefore T(x) = (0,s)?

 

[LCKurtz]

Hmm no I just presumed intuitively that that was the answer, i didn't think it all the way out like that. So for part b, would I write W1 = (0,s), W2 = (s,0) so W1 + W2 = (s,s) and therefore T(x) = (0,s)?

No. Look at the way I illustrated what you should have done for (a.). You have to start with (a,b) and write it as the sum of two points, one from each subspace. It might help to draw a picture.

 

[LCKurtz]

I have to leave now. Another suggestion that might help you is to change way the problem is stated from:

Find a formula for T(a,b) where T represents the projection on the y-axis along the line L={(s,s):s is an element of R}

to

Find a formula for T(a,b) where T represents the projection on the y-axis parallel to the line L={(s,s):s is an element of R}

Project it parallel, that is geometrically what is happening.