72.876 linear-algebra questions.

45 If the field of a vector space weren't characteristic zero, then what would change in the theory?

In the book of Linear Algebra by Werner Greub, whenever we choose a field for our vector spaces, we always choose an arbitrary field $F$ of characteristic zero, but to understand the importance of the ...

Let $u$ and $v$ in $\mathbb R^n$. Evaluate $\|3u - 2v\|$ given that $\|u\| = 4, \|v\| = 5$, and $u\cdot v= 3$

I kind of see the reason why I need dot product in here, but I don't know how to use the dot product to help me figure out this question.

4 How is the derivative of a function $V : \mathbb{R}^{n\times n} \mapsto \mathbb{R}$ defined?

2 answers, 24 views calculus linear-algebra

2 How to show that one linear map is injective and the other surjective when you compose two of them together

If I have $T\colon \mathbb{C}^{3}\to\mathbb{C}^{2}$ and $S\colon \mathbb{C}^{2}\to\mathbb{C}^{3}$ which are both linear maps and we know that $\mathrm{rank}(ST)=2$. How would I show that $T$ is ...

2 Skew-symmetric matrices and invariants

Consider the set of skew-symmetric matrices $AS(n)=\{M \in M(n,n,\mathbb{R}): M^T=-M\}$. How to prove that, to the inner dot product $\left\langle .,. \right\rangle$ of $\mathbb{R^n}$, if $W$ is ...

How many eigenvalues does $f$ have if ${f}^{m}$= Id for some $m$

I found this problem, with $f\in\text{End}_\Bbb{C}(V)$ such that ${f}^{m}= Id_V$ for some $m$. I know that $f$ is diagonalizable because it can't be nilpotent because of the hypothesis. The problem is ...

1 Math and linear algebra prerequisites for convex optimization

I’m trying to learn convex optimization watching Ryan Tibshirani lectures. I sometimes have a hard time understanding basic math in his lectures. Like sets and their properties or how you define a ...

Finding matrix of map, Bilinear form

0 answers, 20 views linear-algebra bilinear-form

-1 Quadratic form using Lagrange methods

I need to find a new base where X1X2+X2X3+X3X4 Is the sum of squares( diagonal matrix) Im trying with Lagrange method but without success (row and column actions also accepted) If anyone can ...

2 Properties of set $\{X∈ M_n(\mathbb{C}) | \operatorname{adj}(X)=I_n\}$

1 answers, 31 views linear-algebra matrices vector-spaces
"For $n≥4$, let $U=\{X∈ M_n(\mathbb{C}) | \operatorname{adj}X=I\}$. Which of the following statements are true/false? Justify: $U$ contains $n-1$ elements, $U$ contains only scalar matrices, if \$X_1,...