Linear Algebra applications

I have this exercise that I'm having trouble with. Just to be clear, this is not a homework assignment, it's just an exercise from my book that will help me understand better what I have to do. So I'm not asking anyone to do a homework in my place, it's just for practice.

From previous exercises, I've already obtained the equations of a plane and a line.

Δ = x + 5y + 5z = 0 (plane)
and the line D:
x = t , y = 3t , z = 5t

C is (i,j,k) from V3
T is the projection on the plane Δ, and parallel to D.

(a) Give A = [T]c
(b) Check that tr(A) = 2 and det(A) = 0
(c) Give a relation between the column of A
(d) What does 2T - id represent^

Thanks a lot to whoever can help me!

Quote:

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Originally Posted by Slyy23

I have this exercise that I'm having trouble with. Just to be clear, this is not a homework assignment, it's just an exercise from my book that will help me understand better what I have to do. So I'm not asking anyone to do a homework in my place, it's just for practice.

From previous exercises, I've already obtained the equations of a plane and a line.

Δ = x + 5y + 5z = 0 (plane)
and
x = t , y = 3t , z = 5t (line)

(a) Give A = [T]c
(b) Check that tr(A) = 2 and det(A) = 0
(c) Give a relation between the column of A
(d) What does 2T - id represent^

Thanks a lot to whoever can help me!

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Hi Slyy23! :)

You do realize that if you want to practice, you should try something and show that?

Anyway, in (a) you introduce the symbols A, T, and c... without explaining what they are supposed to represent.
Without that, we won't be able to give any help on your problems.

Oh yeah sorry I forgot =s I'll edit it. So
C is simply (i,j,k) from V3
and T is the projection on the plane Δ, and parallel to D.

Quote:

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Originally Posted by Slyy23

Oh yeah sorry I forgot =s I'll edit it. So
C is simply (i,j,k) from V3
and T is the projection on the plane Δ, and parallel to D.

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Parallel to D?
What is D?

To construct the matrix A, you will need to pick a couple of independent vectors and figure out what their image is.
Typically you'd select 2 independent vectors that are inside the plane: they will be mapped to themselves.
And the 3rd vector should be this D vector that you just introduced.

Quote:

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Originally Posted by ILikeSerena

Parallel to D?
What is D?

To construct the matrix A, you will need to pick a couple of independent vectors and figure out what their image is.
Typically you'd select 2 independent vectors that are inside the plane: they will be mapped to themselves.
And the 3rd vector should be this D vector that you just introduced.

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Again sorry ._. D is the line mentionned at the beggining. I forgot to name it
x = t , y = 3t , z = 5t (line D)

Quote:

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Originally Posted by Slyy23

Again sorry ._. D is the line mentionned at the beggining. I forgot to name it
x = t , y = 3t , z = 5t (line D)

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Right!
So can you find or pick the 3 vectors I just mentioned?

So here are my two vectors for the plan.

5i + 0j - 1k
0i + 1j - 1k

and for the line I used t=1 to get

i + 3j + 5k

What should I do now?

Not sure if you are still willing to help but I found answers to a) and b) but now I'm stuck on c) and d). So, I found

[T]c =
40/41 -5/41 -5/41
-3/41 26/41 -15/41
-5/41 -25/41 16/41

This works with b) Tr(A) = 2 and det(A) = 0.

Hmm, not sure what you did, but I get something that looks like it, but is still different:



See Wolfram|Alpha.

This also has Tr(A)=2 and det(A)=0

Anyway, for (c) you need to find a linear combination of the columns of A that result in the null vector.
Can you think of a vector that is mapped by T to the null vector?

And for (d) we need to take a look at those 3 vectors that you selected.
What will (2T-I) applied to a vector in the plane be?
And what is (2T-I) applied to the directional vector of your line?