Articles of linear algebra

Divergence as transpose of gradient?

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?

How do you formally prove that rotation is a linear transformation?

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 […]

Proving Distributivity of Matrix Multiplication

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.

diagonalisability of matrix few properties

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.

The calculation of $\dim(U + V + W)$

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 […]

Is arrow notation for vectors “not mathematically mature”?

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 to check if a symmetric $4\times4$ matrix is positive semi-definite?

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?

On the order of elements of $GL(2,q)$?

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 […]

Finding the Jordan canonical form of this upper triangule $3\times3$ matrix

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 […]

Proving replacement theorem?

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 […]