Articles of linear algebra

Confusion on different representations of 2d rotation matrices

When I first learned about 2d rotation matrices I read that you represented your point in your new coordinate system. That is you take the dot product of your vector in its current coordinate system and against the new i and j vector in the rotated coordinate system. $\begin{bmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{bmatrix}p=p^\prime$ The columns represent the new i […]

Proof that $\mathbb{R}$ is not a finite dimensional vector space

How can we prove that the vector space of polynomials in one variable, $\mathbb{R}[x]$ is not finite dimensional?

possible signatures of bilinear form on subspaces

Suppose to have been given the signature of a symmetric bilinear form on a finite dimensional vector space. Is there a general rule to get all the possible signatures of the restriction to subspaces of codimension 1? For instance, I know that if the signature is (-,+,+,+) all subspaces of codimension one have signatures (+,+,+), […]

Existence of $v\in\mathcal{L}(E)$ such as $u=u\circ v\circ u$

Let $E$ be a vector space of dimension $n$, $u\in\mathcal{L}(E)$. How can I show there exists $v\in\mathcal{L}(E)$ such as $u=u\circ v\circ u$ ? $u(x)=0\Rightarrow u(v(u(x)))=0\Rightarrow u(v(0))=0\Rightarrow u(0)=0$ so there isn’t any problem with $\ker$… what now ?

Proof of Non-Convexity

Am looking for a proof of non-convexity of the quotient of two matrix trace functions as given by $\frac{\operatorname{Tr}X^TAX}{\operatorname{Tr}X^TBX}$, when $TrX^TBX>0$ for two different positive semi-definite matrices , $A$ and $B$ such that $A \neq \alpha B$ for any scalar $\alpha$.

Prove $\ker {T^k} \cap {\mathop{\rm Im}\nolimits} {T^k} = \{ 0\}$

Let $V$, a vector space of dimension $n$, and a linear operator $T:V\rightarrow V$. Prove: $\forall k \ge n: \ker {T^k} \cap {\mathop{\rm Im}\nolimits} {T^k} = \{ 0\}$ For a start I chose $v \in$ the intersection. Therefore, $T^k(v) = 0$ and there’s $u \in V$ such that $T^k(u) = v$. What to do next? […]

The link between vectors spaces ($L^2(-\pi, \pi$) and fourier series

So in my PDE course we started with a review of complex numbers and vector spaces to introduce us to fourier series. I have a few questions about this. I know ‘big ell 2’ and ‘little el 2’ are vector spaces. However I need a little bit more understanding on what these are. The lecture […]

Can any linear transformation be represented by a matrix?

Use $\cal L$ to denote a linear transformation on some vector space. We know any matrix $\bf{A}$ can be viewed as a linear transformation by defining $\cal L:= \cal L(\bf{v})= Av$ where $\bf{v}$ is a vector. I am curious is any linear transformation can be represented by a matrix? If so, why? If not, can […]

Singular Value Decomposition of Rank 1 matrix

I am trying to understand singular value decomposition. I get the general definition and how to solve for the singular values of form the SVD of a given matrix however, I came across the following problem and realized that I did not fully understand how SVD works: Let $0\ne u\in \mathbb{R}^{m}$. Determine an SVD for […]

Understanding dot product of continuous functions

I’m reading about Fourier analysis and in my book the author speaks about dot product for continuous functions $f, g\in L^2(a,b)$(the set of functions which are square-integrable on the interval $[a,b]$), which is defined as: $$\langle f, g\rangle = \int_a^b f(x)\overline{g(x)}\;dx.$$ The author mentions that the reader should think the functions as vectors with the […]