Articles of linear algebra

Why $\dim U+\dim U^\perp=\dim V$?

Let $V$ be a vector space. Given $U\le V$, define $U^\perp=\{f\in V^\ast\mid f(u)=0\,,\;\forall u\in U\}$ and given $W\le V^\ast$, define $W^\perp=\{u\in V\mid f(u)=0\;,\;\forall f\in W\}$. This defines a Galois connection between subspaces of $V$ and subspaces of $V^\ast$. If $V$ is finite dimensional, why is $\dim U+\dim U^\perp=\dim V$? Unfortunately, I am not very comfortable […]

Interpolation in $SO(3)$ : different approaches

I am studying rotations and in particular interpolation between 2 matrices $R_1,R_2 \in SO(3)$ which is: find a smooth path between the 2 matrices. I found some slides about it but not yet a good book, I asked the author of the slides and he told me he does not know about a good book […]

Does a matrix represent a bijection

We have a square binary matrix that represents a connection from rows to columns. Is there a way to tell if a bijection exists (other than checking for all possible bijections and iterating through them)? EXAMPLE: 1 0 0 0 0 1 0 0 0 0 1 1 0 0 1 1 ANSWER: Yes, here […]

On integral of a function over a simplex

Help w/the following general calculation and references would be appreciated. Let $ABC$ be a triangle in the plane. Then for any linear function of two variables $u$. $$ \int_{\triangle}|\nabla u|^2=\gamma_{AB}(u(A)-u(B))^2+ \gamma_{AC}(u(A)-u(C))^2+\gamma_{BC}(u(B)-u(C))^2, $$ where $$ \gamma_{AB}=\frac{1}{2}\cot(\angle C), \gamma_{AC}=\frac{1}{2}\cot(\angle B), \gamma_{BC}=\frac{1}{2}\cot(\angle A). $$ What is a good reference for the formula? Is it due to R. Duffin? […]

Determining similarity between paths (sets of ordered coordinates).

With limited knowledge of mathematics, I am not sure what tags to use for this question. I have a path on a 2D surface called $(p1)$. A path consists of a set of ordered $(x,y)$ coordinates. By ordered I mean the first line segment in a path would be $(x1,y1) to (x2,y2)$, the second line […]

Composition of linear maps

I have question as follows: Let $V$ be a vector space of dimension 3 over $\Bbb R $ and let t $\in$ $\mathcal L (V,V)$ have eigenvalue $-2,1,2$ . Use the Calyey-Hamilton theorem to find $t^4$ I know that $(t-2)(t-1)(t+2) =0$ so $t^3 = t^2 +4t -4id$ I am unsure how to proceed

Elementary proof that $Gl_n(\mathbb R)$ and $Gl_m(R)$ are homeomorphic iff $n=m$

This question already has an answer here: Elementary proof that $\mathbb{R}^n$ is not homeomorphic to $\mathbb{R}^m$ 6 answers

References for coordinate-free linear algebra books

I’m looking for some (or one) good book(s) that teach linear algebra either purely coordinate-free or ones that present the standard bag-of-tools alongside coordinate-free alternatives or discussions. Thanks!

Showing the polynomials form a Gröbner basis

Let $A$ be an $m \times n$ real matrix in row echelon form and $I \subset \mathbb{R}[x_1,\dots,x_n]$ is an ideal generated by polynomials $p_i = \sum_{j = 1}^na_{ij}x_j$ with $1 \leq i \leq m$. Then the generators form a Gröbner basis for $I$ w.r.t. some monomial order. So I guess one should try the standard […]

Does linear dependency have anything to do when determining a span?

Q: Does $\{(1,1) , (2,2)\}$ span $\mathbb{R}^2$? A: No, because they are linearly dependent. I agree that it doesn’t span $\mathbb{R}^2$, but from my understanding, linear dependency has nothing to do with that: All that matters is whether you are capable of producing any vector in $\mathbb{R}^2$ by some sort of linear combination of the […]