Question

I have a question regarding exercise 14 in chapter 4.4 of Strang's book. It is clear from the theory part that the missing element in the 2x2 matrix should be equal to q1^T*b. But when I try to work backwards through Gram-Schmidt, i don't get this answer. Probably something very silly, but i cannot figure out what i am doing wrong. To get back to vector b, i need to take a certain combinations of the columns of Q. That is: b = ?*q1+||B||*q_2. This gives b = ?*q1+B. From the first step of Gram-Schmidt we have b = B + (a^Tb)/(a^Ta)a, therefore ? ends up being equal (a^T*b).

Answer 1

You've made a tiny error:
From the first step of Gram-Schmidt we have
\[b = B + \frac{A^Tb}{A^TA}A\]
that's right.

But if you plug it back in your equation
\[
\begin{align*}
b &= ?q_1+ B\\
B + \frac{A^Tb}{A^TA}A &= ?q_1+ B\\
\frac{A^Tb}{A^TA}A &= ?q_1\\
A^Tb\frac{A}{||A||^2} &= ?\frac{A}{||A||}
\end{align*}
\]

you can see that \(?\) has to be \(\frac{A^Tb}{||A||} = \frac{A}{||A||}^Tb\) which is \(q_1^Tb\).

Answer 2

Thanks! I know exactly what got me confused. The capital letters that are used to express vectors.
\[A ^{T}A\]
read as a matrix product to me. Damn this notation! :-)