Question

Find the perpendicular projection of (1,1,1) on the one dimensional subspace of C^3 spanned by(1,-1,1). In other words, find the image of the given vector under the projection onto the given subspace
Please, . I need guidance.

Answer 1

@tkhunny

Answer 2

I might start with a Basis. Do you have one of those?

Answer 3

Then, you'll need a definition for an Inner Product.

Answer 4

@Spacelimbus please, tell me what should I do for this problem

Answer 5

shame on me, I even not know how to start

Answer 6

I know the projection mappings, but sure if they're asking for this though, the wording is a bit complex to me at the moment :)

Answer 7

I think the given vector is in standard basis, and the given subspace is in another basis. Do I have to change the basis first by construct an orthonomal basis from (1,-1,1)?

Answer 8

Normally you don't, but to construct an ONB you'd need to have a scalar product defined (bilinear, symmetric, positive definite mapping), otherwise the idea of orthogonality does not make much sense.

Answer 9

:( !!

Answer 10

but for the sake of the exercise I presume they just want you to use the standard scalar product (dot product/ inner product).

Answer 11

it is in inner product part, so that I can assume that they want me to use inner product form

Answer 12

Okay, Standard Inner Product. That's where we started. \(\dfrac{(\alpha|\beta)}{(\beta|\beta)}\beta\) for \(\alpha = (1,1,1)\;and\;\beta = (1,-1,1)\). Notice how \(\dfrac{\beta}{(\beta|\beta)}\) IS the Orthonormal Basis Vector.

Answer 13

I am sorry, I don't get the form. Is it \(\dfrac{<\alpha,\beta>}{||\beta||^2}\beta\) ?

Answer 14

Let M ={(m1,m2,m3) such that m2 = -m1, and m1 =m3}
so, M ={\( (c, -c, c)^\perp; c\in C\)}
find \(M^\perp \) = {\((c, -c, c); c \in C\)}

Answer 15

Same thing.

Answer 16

I understand that no one owes me anything. However, I wonder how the stuff helps? for the very abstract problem like this, even the helper types the steps explicitly, the asker needs a lot of trying to understand what happen still. If the helper gives out just 1 sentence /time, the asker has to spend 1year to solve 1 problem and he (the asker) will die before he gets the final answer. :) :] :( :( :> :-> :-) :) :)

Answer 17

Not every forum or venue is particularly useful for every question in ever subject or for every student. Some things require a more active discussion. Some things work fine with just a bulletin board. This forum is a hybrid between those two.

Answer 18

Rich + Not willing to spend = Poor