- #1

- 148

- 1

## Homework Statement

Let

*A*be a

*m*x

*n*matrix of rank

*n*and let [tex]\textbf{b} \in R^{m}[/tex]. If

*Q*and

*R*are the matrices derived from applying the Gram-Schmidt process to the column vectors of

*A*and

p =

*c*

_{1}

**q**

_{1}+

*c*

_{2}

**q**

_{2}+ ... +

*c*

_{n}**q**

_{n}is the projection of

**b**onto

*R(A)*, then show that:

a) c =

*Q*

^{T}

**b**

b) p =

^{T}

**b**

c)

^{T}=

*A*(

*A*

^{T}

*A*)

^{-1}

*A*

^{T}

## Homework Equations

I'm not quite sure about the (a).

But in (b) I have, from my book:

*Let*

*S*be a nonzero subspace of*R*and let [tex]\textbf{b} \in R^{m}[/tex]. If {^{m}**u**_{1},**u**_{2},...,**u**_{k}} is an orthonormal basis for*S*and*U*= (**u**_{1},**u**_{2},...,**u**_{k}), then the projection**p**of**b**onto*S*is given by**p**=

*UU*

^{T}

**b**

In (c) I have (Some place in my book) that:

[tex]\textbf{p} = A\hat{x} = A\left(A^{T}A\right)^{-1}A^{T}\textbf{}[/tex] which is called the projection matrix.

## The Attempt at a Solution

As I said, I don't know about (a), but in (b) my thought was that the projection

**p**of

**b**is given by:

**p**=

*c*

_{1}

**u**

_{1}+ ... +

*c*

_{k}

**u**

_{k}=

*U*

**c**, which is pretty much the same as given in the problem, and

**c**= (c

_{1}, c

_{2},...,c

_{k}) = (

**u**

_{1}

^{T}

**b**,

**u**

_{2}

^{T}

**b**,...,

**u**

_{k}

^{T}

**b**) = U

^{T}

**b**

and then I get:

**p**=

*UU*

^{T}

**b**, which is what I needed to find (except for some other letters used in this problem).

In (c) we have the p from (b) which were

**p**=

*UU*

^{T}

**b**. So by applying

**p**on

**p**'s place in the equation given in "Rel. eq" I get

*UU*

^{T}

**b**= [tex]A\hat{x} = A\left(A^{T}A\right)^{-1}A^{T}\textbf{}[/tex] which is pretty much the thing I needed to show, except for the two

**b**'s on each side.

But I don't know if I just can remove them, just if they were numbers? Because that would give me the correct equation.

Hope I haven't made it to confusing. At least I hope for some hints for (a), and maybe a hint about (b) and (c)'s correctness ? :)

Regards