In his online lectures on Computational Science, Prof. Gilbert Strang often interprets divergence as the “transpose” of the gradient, for example here (at 32:30), however he does not explain the reason. How is it that the divergence can be interpreted as the transpose of the gradient?

The fact that rotation about an angle is a linear transformation is both important (for example, this is used to prove the sine/cosine angle addition formulas; see How can I understand and prove the "sum and difference formulas" in trigonometry?) and somewhat intuitive geometrically. However, even if this fact seems fairly obvious (at least from […]

If $A,B,C$ are matrices I am thinking how to show that $$ A(B + C) = AB + AC$$ Is possible to show without sums like $\sum_i a_i, …, \sum_j b_j$? It seems if I do the proof with many indexes then is tedious and I don’t learn much from it.

What are the most important things one should remember to check the diagonalisability of a matrix? Please help, I have exams on next week. Say some best and easy methods,time efficient.

Is there a (valid) formula for $\dim(U + V + W)$? I know from MO that $$\begin{align*} \dim(U + V + W) &= \dim(U) + \dim(V) + \dim(W)\\ &\qquad\mathop{-} \dim(U \cap V) – \dim(U \cap W) – \dim(V \cap W)\\&\qquad \mathop{+} \dim(U \cap V \cap W) \end{align*}$$ is wrong. Can we relate $\dim(U + V […]

Assuming that we can’t bold our variables (say, we’re writing math as opposed to typing it), is it “not mathematically mature” to put an arrow over a vector? I ask this because in my linear algebra class, my professor never used arrow notation, so sometimes it wasn’t obvious between distinguishing a scalar and a vector. […]

How does one check whether symmetric $4\times4$ matrix is positive semi-definite? What if this matrix has also rank deficiency: is it rank 3?

There’s a particular property of the elements of $GL(2,q)$, the general linear group of $n\times n$ matrices over a finite field of order $q$, that I don’t understand. First, I know that the order of $GL(2,q)$ is $(q^2-1)(q^2-q)=q(q+1)(q-1)^2$, since there are $q^2-1$ possible vectors for the first column, excluding the $0$ vector, and $q^2-q$ possible […]

I am supposed to find the Jordan canonical form of a couple of matrices, but I was absent for a few lectures. \begin{bmatrix} 1 & 1 & 0 \\ 0 & 1 & 2 \\ 0 & 0 & 3 \end{bmatrix} Since this is an upper triangular matrix, its eigenvalues are the diagonal entries. Hence […]

I want to see if I am understanding the proof of the replacement theorem correctly. Let $V$ be a vector space that is spanned by a set $G$ containing $n$ vectors. Let $L \subseteq V$ be a linearly independent subset containing $m$ vectors. Then $m\leq n$ and there exists a subset H of G containing […]

Intereting Posts

Real-world applications of prime numbers?
Peano arithmetic with the second-order induction axiom
Could I be using proof by contradiction too much?
For any integer n greater than 1, $4^n+n^4$ is never a prime number.
Geometric series of matrices
$1+a$ and $1-a$ in a ring are invertible if $a$ is nilpotent
Inverse Mellin transform of $f(s)= 2^{ \frac{s}{6} }\frac{\Gamma \left( \frac{s+1}{3/2} \right)}{ \Gamma \left( \frac{s+1}{2} \right)}$
Bounds on the gaps in a variant of polylog-smooth numbers.
What is the expected number of trials until x successes?
If $ \mathrm{Tr}(M^k) = \mathrm{Tr}(N^k)$ for all $1\leq k \leq n$ then how do we show the $M$ and $N$ have the same eigenvalues?
$R\subseteq S$ integral extension and $S$ Noetherian implies $R$ Noetherian?
Why do we use this definition of “algebraic integer”?
Every power series expansion for an entire function converges everywhere
Prove that $R$ is a commutative ring if $x^3=x$
Showing any countable, dense, linear ordering is isomorphic to a subset of $\mathbb{Q}$