Finding orthonormal set using Gram-Schmidt and least squares

[tatianaiistb]

Homework Statement

A) Find an orthonormal set q1, q2, q3 for which q1,q2 span the column space A [1 1; 2, -1; -2,4] (this is a 3x2 matrix).

B) Which fundamental subspace contains q3?

C) What is the least-squares solution of Ax=b if b=[1 2 7]^T?

Homework Equations

Gram-Schmidt

The Attempt at a Solution

A) So I figured out q1, q2, and q3 using Gram-Schmidt process. These are q1 = [1/3 2/3 -2/3]^T, q2 = [2/3 1/3 2/3]^T, and q3 = +-[-2/3 2/3 1/3]^T

B) Don't know how to determine this

C) For the least square, since they are orthonormal sets, I used the formula x= [q1^Tb; q2^Tb] However, I'm not getting the right answer. In addition, I tried doing it the long way using the formula x=(A^TA)^-1A^Tb, I'm getting even different numbers... Please help!

 

[TheoMcCloskey]

You found the orthonormal set (Q), but that is not equal to A. What is A equal to?

 

[tatianaiistb]

[1 1
2 -1 = A
-2 4]

I'm now only having trouble with part C. Thanks!

 

[TheoMcCloskey]

Yes, I understand. But there is another way to represent $A$ using the orthogonal basis that you constructed and another factor of $A$. This factor, call it $R$, is an upper right triangular matrix. $R$ is constructed such that $QR = A$. We note $Q$ is orthogonal so $Q^{T} = Q$. Thus, using this factorization for $A$ in the normal equations yields a linear triangualr system for solving the Least Squares problem.

 

[TheoMcCloskey]

I should add that, as part of the construction of Q, you will have already computed the elements of R.

 

[tatianaiistb]

Thanks Theo! I do understand about A=QR and the construction of it. But I don't understand how that can help me finding x(hat), which is the least-squares solution.

What I was thinking is that x(hat)=Q^T*b because Q has already orthonormal vectors.

I'm getting that x(hat) = (-3 6 3)^T, but it says the solution is supposed to be x(hat) = [1 2]^T which they supposedly got from q1^Tb and q2^Tb. However, if I multiply the very same q's I got (which I was confirmed are correct) by the given b, I'm not getting the given answer. So I'm going in circles... Any help?

 

[TheoMcCloskey]

Firstly, let me make a correction to one of my previous posts (post #4).

We note $Q$ is orthogonal so $Q^{T}=Q$.

should read

We note $Q$ is orthogonal so $Q^{T}=Q^{-1}$.

Secondly, return to the Normal equations for the Least Squares solution:

$$A^{T}A \, x = A^{T} \, b$$
Now substitute the factor expresion for $A$, ie, $A=QR$
$$(QR)^{T}(QR)\,x = (QR)^{T}b$$
$$R^{T}(Q^{T}Q)\,R\,x = R^{T}Q^{T}b$$
$$R^{T}R\,x = R^{T}Q^{T}b$$
since $Q^{T}Q=I$. Now pre-multiply both sides by $R^{-T}$ to yield
$$R\,x = Q^{T}b$$
It is this system that you need to solve for $x$ to yield the Least Squares solution. Remember, just because you found the orthogonal basis for $A$ doesn't mean its equal to $A$ . There is a certain amount of translation and rotation of that basis to yield that $A$ (hence the $R$ factor).

Remember, $R$ is triangular, so a simple backsolve operation will yield the solution (no need to compute $R^{-1}$).