# Introduction to Linear Algebra (Strang) 2.7 Example 3

Image of Problem: https://i.sstatic.net/R8TiR.png

I have been working through Gilbert Strang's Introduction to Linear Algebra, but I don't understand a step of Section 2.7 Example 3. Strang compares the "difference matrix" {-1,1,0; 0,-1,1} with the derivative operator $d/dt$. He argues that the transpose of the difference matrix is equivalent to $-d/dt$, that the derivative is "anti-symmetric."

I understand this idea in theory, but his justification of this anti-symmetry is done using integration by parts, as seen in the image. In my understanding, integration by parts is derived via the product rule:

$\frac{d}{dt}(f(t) g(t)) = f'(t) g(t) + f(t) g'(t)$

Strang's explanation, however, ignores the left side of this equation. Why is this?

The left-hand side of the equation when integrated $-\infty$ to $\infty$ gives $$\left.f(t)g(t)\right|_{t = -\infty}^\infty$$ so only depends on the boundary and is often called the "boundary term". Often it's assumed that the functions decay as $|t|\to \infty$ so this term is zero.