I am trying to teach my self some linear algebra in preparation for a module in machine learning. I am using Gilbert Strang's text Introduction to Linear Algebra and am having some difficulties.
My specific question is: How is the last equation below an example of integration by parts? It seems to be missing $x(t)y(t)$ to me and I don't know where that has gone.
The book gives the following equations leading up to it:
$$x^Ty = (x,y) = \int_{-\infty}^\infty x(t)y(t)~dt$$
From what I can tell, the above is saying that the inner product of the vectors $x$ and $y$ is equivalent or at least approximate to taking the integral of those two vectors as functions. Does the $(x,y)$ just mean inner product? I don't really understand what a vector of a function even is though.
$$(Ax)^Ty = x^T(A^Ty)$$
This associative rule for matrix-vector multiplication was given as a more rigorous view of what the transpose of $A$ actually is. It then partnered it with this:
$$\int_{-\infty}^\infty\frac{dx}{dt}y(t)~dt = \int_{-\infty}^\infty x(t)\left(-\frac{dy}{dt}\right)~dt$$
I can see how the two equations relate to eachother and that this suggests that $A^T$ is anti-symmetric. Gil mentions integration by parts here which I can definitely recognise but it doesn't seem complete, there should be an extra $x(t)y(t)$ surely?
It seems that you are somewhat confused about the use of "vectors" terminology. In the context as mentioned, vectors are quite abstract things. In fact, they are a generalization of the well known "geometric entities endowed with magnitude and direction". See the Wikipedia page about Vector spaces , especially the section about Function spaces .
So there is not a "vector of a function": the functions themselves are abstract vectors. And yes, an inner product - abstract again - for functions can be defined by the integral as mentioned. What must been shown, however, is that the thus defined integral preserves some essential properties of the common "geometric" inner product. See the Wikipedia page about Inner product spaces .
The main part of your question ( i.e. why $\left[x(t)y(t)\right]_{-\infty}^{+\infty} = 0$ ) has been answered by 'user161825' .
Hope this helps.