Questions tagged [linear-algebra]

Question 1

I'm starting my linear algebra studies and came across the following statemtent: $E = F(\mathbb{R};\mathbb{R})$ is the vector space of the one variable real functions $f:\mathbb{R} \rightarrow \...

Question 2

I am trying to understand definition of Euclidean norm on a finite dimensional space, $\mathbb{R}^{n}$, denoted as $\mathbb{E}$. The dual space is denoted by $\mathbb{E}^{*}$. In the attached ...

Question 3

Let $V,W$ be (finite dimensional) vector spaces over a field $\mathbb{K}$. Construct the tensor product between them as the quotient $\mathcal{F}(V\times W)/R$ where $\mathcal{F}(V\times W)$ is the ...

Question 4

Given a unit vector $u$ and another unit vector $v$, I want to rotate $u$ into $v$ in two stages. In the first stage, I rotate $u$ about a given axis $a_1$ (by an unknown angle) to produce a vector $...

Question 5

Or equivalently say, suppose $V$ is a vector space, any two bases of $V$ have the same cardinality?

Question 6

For the case of commutative field, we have that each vector space has a basis and even more, each linearly independent set can be completed into a basis. Do we have a same result for general ...

Question 7

Let $w_1,\dots,w_k$ be linearly independent and $W=\mathrm{span}\{w_1,\dots,w_k\}$. If $w\in W$, then $$ w = a_1 w_1 + \cdots + a_k w_k $$ for some scalars $a_i$, then $\{w,w_1,\dots,w_k\}$ is l.d. ...

Question 8

In the book of sakurai Modern Quantum Mechanics they have this $$ \begin{aligned} & \sqrt{(j \mp m)(j \pm m+1)}\left\langle j_1 j_2 ; m_1 m_2 \mid j_1 j_2 ; j, m \pm 1\right\rangle \\ & =\sqrt{...

Question 9

Let $L_S:F^n→F^m$ be a linear map which has null space $O$. Prove: if $H^1,\dots,H^k$ are linearly independent elements in $F^n$, then $L_S (H^1 ),\dots,L_S (H^k )$ are linearly independent elements ...

Question 10

The linear map $\mathcal{A}(\cdot)$ is said to have RIP (restricted isometry property) with a restricted isometric constant ${\delta\in [0,1)}$ if it has the following property: \begin{equation}\label{...

Question 11

Let $A = [a_{ij}]_{n \times n}$ and suppose the minimal polynomial is $ m_A(x) = (x-2)^4 (x-1)^4 $ How many Jordan canonical forms are possible? Since the exponent $4$ forces the largest Jordan block ...

Question 12

I have this question on a linear algebra worksheet Let $V$ be the vector space of functions from $\mathbb{R}$ to $\mathbb{R}$. Show that $f,g,h \in V$ are linearly independent, where $f(t)=\sin(t)$, $...

Question 13

In the book of sakurai Modern Quantum Mechanics they have this $$ \begin{aligned} & \sqrt{(j \mp m)(j \pm m+1)}\left\langle j_1 j_2 ; m_1 m_2 \mid j_1 j_2 ; j, m \pm 1\right\rangle \\ & =\sqrt{...

Question 14

We will use Einstein summation convention. I apologize if this question is too easy—it has been years since I've had to to work with some of the mentioned material. Have the the set of all linear ...

Question 15

I am currently learning about linear algebra. To better understand linear algebra and create a network of mathematical concepts, I would like to know what the Big Picture of Linear Algebra is. Which ...

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Questions tagged [linear-algebra]

How to prove that $C^k(\mathbb{R})$ is a subspace of $F(\mathbb{R};\mathbb{R})$

Using self-adjoint operator to define Euclidean norm

Prove $V\otimes W \cong \mathcal{L}(V^,W^;\mathbb{K})$

Rotating a unit vector to another vector using two consecutive axes

Is the "cardinality of the dimension of a vector space with basis" well-defined? [duplicate]

Semivector spaces over commutative semifields

Show that $w\in W$ if and only if $w\wedge w_1\wedge\cdots\wedge w_k = 0$ in $\wedge^{k+1}V$

Proportionality of solution of a system of equation

Proof of Linearly independece set [duplicate]

How do RIP constants change when the constraint set is enlarged?

Possible number of Jordan canonical forms

Doubt on a proof that $\sin(t)$, $\cos(t)$, and $t$ are linearly independent in the vector space of functions from $\mathbb{R}$ to $\mathbb{R}$

If 2 recursion relations have the same coefficients, are the solutions proportional?

Does the tensor $\Phi$ take this form?

What is the Big Picture of Linear Algebra? [closed]

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