Articles of linear algebra

Understanding a Proof for Why $\ell^2$ is Complete

Setting: Let $(x_n)$ be Cauchy in $\ell^2$ over $\mathbb{F} = \mathbb{C}$ or $\mathbb{R}$. I’m trying to show that $(x_n) \rightarrow x \in \ell^2$. That is, I’m trying to show that $\ell^2$ is complete in a particular way outlined below. I only used the first few steps of the proof because once I understand the third […]

Using Linear Transformations to Solve Differential Equation

Define $T:P_3 -> P_3$ by $T(f)(t) = 2f(t)+(1-t)f'(t)$. a) Show that T is a linear transformation. b) Give the matrix representing T with respect to the “standard basis” {$1,t,t^2,t^3$}. c) Determine ker(T) and Image(T). d) Let $g(t) = 1+2t$. Use your answer from (b) to find a solution to the differential eqn $T(f)=g$. e) What […]

What should be the characteristic polynomial for $A^{-1}$ and adj$A$ if the characteristic polynomial of $A$ be given?

Let the characteristic polynomial of $A$ be $\psi_A(x):=p(x)$. If $A$ be non-singular, then find that the characteristic polynomial of $A^{-1}$ and adj$(A)$. My attempt: We have \begin{align*} &\psi_{A^{-1}}(x)\\ =&|xI_n-A^{-1}|\\ =&|A^{-1}||xA-I_n|\\ =&|A|^{-1}x^n |A-\frac 1x I_n|\\ =&(-1)^nx^n|A|^{-1}\psi_A(x)\\ =&(-1)^nx^n|A|^{-1} p(x) \end{align*} In this way, we can find the characteristic polynomial for $A^{-1}$ in terms of the characteristic polynomial […]

Sign of det(UV) in SVD

Let $A\in \mathbb{R}^{n\times n}$. Its Singular Value Decomposition (SVD) is $$A=U\Sigma V^T$$ We know $U$ and $V$ are orthogonal matrices. Sometimes $\det UV=1$ and sometimes $\det UV=-1$. My question is: what kind of matrices give $\det UV=1$? Can we say something about the sign of $\det UV$ based on some properties of $A$ before we […]

Weights – Objects into bags puzzle

I found a maths puzzle somewhere and a part of it as below: Kelly wants to place n objects $a_1 , a_2 , ··· , a_n$ into $k > 1$ bags. For each $i = 1 , 2 , ··· , n $, the weight of $a_i$ is $w_i$ kilograms. The capacity of each bag […]

Reconstructing a Matrix in $\Bbb{R}^3$ space with $3$ eigenvalues, from matrices in $\Bbb{R}^2$

I have a matrix which represents a closed loop matrix of a control system with delays (Control Systems Theory) in $\Bbb{R}^3$ space that has $3$ eigenvalues. Through some process I have obtained three different matrices in $\Bbb{R}^2$ and sometimes in $\Bbb{C}^2$ which represent a part of the control system and each matrix has just two […]

Warm start of simplex algorithm after update of constraint matrix

Assume we found an optimal solution $\mathbf{x}_1$ of the linear program \begin{gather} \max \mathbf{n}^T\mathbf{x}\mbox{ s.t. }A\mathbf{x} \leq \mathbf{b}\tag{1} \end{gather} using the simplex algorithm. Now we update the matrix $A$ by a bijective mapping represented by the matrix $M$. Then clearly $\mathbf{y} = M\mathbf{x}_1$ is a (primal) feasible solution of the linear program \begin{gather} \max \mathbf{n}^T\mathbf{x}\mbox{ […]

Isomorphism with dual space

Let $V$ be a vector space over $F$ with basis $\{v_1,\cdots, v_n\}$, and for every $i$, let $f_{v_i}\colon V\rightarrow F$ be a linear map satisfying $f_{v_i}(v_j)=\delta_{ij}$. Then $\{f_{v_1}, \cdots, f_{v_n}\}$ forms a basis of the dual space $V^*$. The map $T\colon v_i\mapsto f_{v_i}$ uniquely extends to a linear injective map from $V$ to $V^*$, and […]

Condition for mapping linearly independent vectors to linearly independent vectors

Let $f \colon U \to V$ be a linear map and $\{u_1, u_2, \ldots, u_n\}$ be the set of linearly independent vectors in $U$. Then the set $\{f(u_1), f(u_2),\ldots,f(u_n)\}$ is linearly independent iff a. $f$ is one-one & onto b. $f$ is one-one c. $f$ is onto d. $U = V$ I came to conclusion […]

difference between parallel and orthogonal projection

i would like to understand what is a difference between parallel and orthogonal projection?let us consider following picture we have two non othogonal basis and vector A with coordinates($7$,$2$),i would like to find parallel projection of this vector to these basis,i am studying Covariant and Contrivant components,so i would like to understand how to find […]