multiplying both sides of a linear equation

Question

I am working through the book "Linear Algebra and Its Applications" By Gilbert Strang and I have come across an equation on page 176 which I cannot get my head around.

This is dealing with orthonormal matrices $Q$ and the equation $Qx = b$

write b as a combination $b = x_1q_1 + x_2q_2 + ... + x_nq_n$

To compute $x_1$ there is a neat trick Multiply both sides of the equation by $q_1^T$. On the left-hand side is $q_1^Tb$. On the right-hand side all terms disappear (because $q_1^Tq_j = 0$ except for the first term. We are left with

$$ q_1^Tb = x_1q_1^Tq_1 $$

and since $q_1^Tq_1 = 1$

$$ x_1 = q_1^Tb $$

My question is about the order of the terms on the right side of the equation. Since order matters, wouldn't it have to be written $q_1^Tx_1q_1$ since the multiplication on the left side was inserted from the left?

Answer 1

$x_1$ is just a scalar, so $q_1^Tx_1q_1=x_1q_1^Tq_1$.

Answer 2

I will assume that $x=(x_1,x_2,\dots,x_n)^t$ and that $q_j$ is the $j$-th column of $Q$. So, in the equation $$Qx=x_1q_1+x_2q_2+\cdots+x_nq_n$$ we have a linear combination of the vectors $q_1,q_2,\dots,q_n$ with coefficients $x_1,x_2,\dots,x_n$. Therefore, for all $j$ we have $$q_1^t (x_jq_j)=x_j(q_1^tq_j)$$ ($x_j$ is just a scalar, more precisely, the $j$-th component of $x$)