Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them. This includes: systems of linear equations, basis, dimension, subspaces, matrices, determinant, trace, eigenvalues and eigenvectors, diagonalization, Jordan form, etc. For questions specifically ...

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Linear functional on complex vector space

y(x) : C --> R is not a linear functional because C is a complex vector space. I do not know why. Please explain me. I can see they are not linear functional by checking only. Is there any logic on ...
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26 views

Easy condition on a matrix $A$ such that $Au\geq 0$ for all vectors $u\geq 0$

A rather easy question, but my brain stopped working. I have a matrix $A$ that depends on various parameters in a non-trivial way. What would be a simple-to-check conditions on $A$ so that $Au\geq 0$ ...
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3answers
79 views

Uniform convergence of matrix integral sequence

I was given recursively defined: $$ M_k(t)=I+\int_{t_0}^tA(s)M_{k-1}(s)~ds $$ and $M_0=I$ and that $A(t)$ is a matrix with entries that are continuous functions on $t_0\leq t\leq t_1$. By induction ...
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65 views

Algorithm to determine matrix equivalence

I'm a physicist who's not particularly good at linear algebra so please accept my apologies if this is standard textbook stuff that I'm just unaware of. I have two real rectangular matrices $A_{mxn} ...
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60 views

Partitioned matrix of partitioned matrices

Please, help solve this question: Given the partitioned matrix \begin{equation} P=\left( \begin{array} {c,c} A \quad B \\ C \quad 0 \end{array} \right) \end{equation} where A is a 2x2 block matrix, ...
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101 views

Calculate the series: $\sum^\infty_{n=1}\frac{(-1)^n}{n^2}$ using dirichlet's theorem

This question was in my exam: Calculate the series: $$\sum^\infty_{n=1}\frac{(-1)^n}{n^2}$$. I answered wrong and the teacher noted: ...
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1answer
41 views

What constraints do we get on the matrices $A,B$ when we require $AV=VB$?

The matrices $A$ and $B$ are, a priori, general unitary $3\times3$ matrices and $V$ is some fixed unitary $3\times3$ matrix. When I impose the following requirement on $A$ and $B$: \begin{equation} AV ...
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89 views

Let A a diagonalizable matrix. Show that $A$ and $A^{t}$ are similar.

Let A a diagonalizable matrix. Show that $A$ and $A^{t}$ are similar.
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96 views

Finding a basic in subspace in vectors space $\mathbb{R}_3[x]$

In vectors space $\mathbb{R}_3[x]$ we got subspace: $U =$ { p $\in$ $\mathbb{R}_3[x]$; p(1) = p'(1)} and $V =$ {p $\in$ $\mathbb{R}_3[x]$; p(1) = $ \int_0^1 p(t)\,dt. $} How can i find basis of ...
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45 views

How to prove this implication?

Let $A$ be a matrix $n \times n$ and $b_1....b_k$ are $k$ vectors in $\mathbb{R}^{n}$. Does anyone know how to prove the following implication $Ab_1, ..., Ab_{k}$ is a spanning set of ...
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40 views

Summing a particular product of binomial coefficients

I expect this is elementary, but I can't find a closed form. Let $a_i$, $i=1,...,m$, be a sequence of natural numbers and $n>\sum a_i$. What is the value of the sum: ...
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186 views

Distance or Similarity between matrices that are not the same size

I have many matrices that have different size. Specifically, those matrices have the same number of rows but vary in the number of column. In another word, I have matrices $A_1,\dots,A_n$ where ...
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1answer
39 views

Proving that the union of two vector sub spaces of the same space are not a sub space. [duplicate]

I hope that I translated it correctly, correct me if I'm wrong :) Let $V_1$ and $V_2$ be sub spaces of $L$. Prove that $V_1\cup V_2$ is a vector sub space if and only if $V_1 \subseteq V_2$ or ...
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79 views

Prove that the operator norm is a norm

Exercise: Prove that the operator norm of the set $S$ of all linear operators $L:R^n\to R^m$ defines a norm on $S$ Definition of norm: A positive function $\| .\|$ on a real vector space $V$ is a ...
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Conditions of a Monotonic Process?

$f$ is the output of a discrete time process described by $f(k)=\sum_{i=1}^{k-1}w_{ki}f(i)$ where $f(1)\geq0$ is a known initial condition and $w_{ki}\geq0$ are weights of previous states on the ...
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2answers
46 views

What is the nullity of an onto transformation?

For a $5 \times 13$ matrix, with $T(x) = Ax$, what is the nullity of $A$ if $T$ is onto? I can't figure out what it would be...
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53 views

Linear Algebra Quiz : 2 Sums

Sum 1: I have 60 more 1 Dollar coins that 5 Dollar coins. The total value of money = $360. How many 1 ans 5 Coins Do I have ? Sum 2: Sum Of the digits of a two-digit number is 15. ...
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31 views

Linear algebra, eigenspace property

Let $\vec{v}$ be an eigenvector of $A$ with corresponding eigenvalue $\lambda$ and $c$ a scalar. show that $v$ is an eigenvector of $A-cI$ with corresponding eigenvalue $\lambda-c$. I'm not sure ...
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1answer
64 views

Are orthogonal spaces exhaustive, i.e. is every vector in either the column space or its orthogonal complement?

Quick question about subspaces, just to make sure I have this straight in my head. Taking an $n\times k$ matrix X with $rank(X)=k$, is every vector in $\mathbb{R}^n$ in either the column space $C(X)$ ...
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36 views

Basis of this vector space.?

Let F denotes the set of sequences such that $u_{n}+u_{n+1}-u_{n+2}=0$. How to find a basis of this vector space? Thanks.
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What is the proper mathematical terminology to use when describing conversion of a black contour to a heat-mapped contour representing curvature?

I have an image processing problem in which I want to take a contour and instead of the contour being represented in black and white I want it represented in a heatmap of the contour's curvature. How ...
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0answers
108 views

mathematics of chemical stoichiometry

I would like to better understand the mathematical description of chemical stoichiometry and thermodynamic chemical equilibrium. This problem has many features and I know my description might be too ...
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0answers
51 views

Proof by contradiction: $E_1+E_2\doteq E_1 \oplus E_2 \leftrightarrow E_1 \cap E_2=\{0_V\}$

I must proof the following: Prop.: Let $E_1,E_2$ two vector subspace of $V$ then $$E_1+E_2\doteq E_1 \oplus E_2 \leftrightarrow E_1 \cap E_2=\{0_V\}$$ Proof: I must show $$1)E_1+E_2\doteq E_1 \oplus ...
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30 views

Line in vector form?

Given the line y=3x my book states it is $\left(\begin{array}{c}1 \\ 3\\\end{array}\right)$ as a matrix. Why is it not $\left(\begin{array}{c}3 \\ 1\\\end{array}\right)$, I thought the upper number ...
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2answers
35 views

Show inequality for two elements in $\mathbb{R}^n$

I know that $x,y\in \mathbb{R}^n$ are such that $x_1\leq0,x_1^2\geq x_2^2+\dots+x_n^2$ and $y_1\geq 0,y_1^2\geq y_2^2+\dots+y_n^2$. Is it possible to show that $$x_1y_1+x_2y_2+\dots +x_ny_n\leq 0$$ ...
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1answer
32 views

Making sure I'm not doing something wrong

I'm solving a matrix equation: $$2(A-B+X) = 3(X-A)$$ Where $A = \left( \begin{matrix} 1&2 \\ 3&4 \end{matrix} \right)$ and $B = \left( \begin{matrix} -1&0 \\ 1&1 \end{matrix} \right) ...
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1answer
45 views

Reduction to Hessenberg-Triangular Form

I am trying to reduce the following pair of matrices into Hessenberg-Triangular form where A is upper hessenberg and B is upper triangular. A =\begin{array}{cc} 1 & 2 & 3 \\ 1 & 3 & 4 ...
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35 views

linear transformation question. where did i go wrong?

i'm sorry if you can't read some parts so ask me if you need to know what it is but i really am confused as to what i did wrong. Here is the picture of the question and answer that i gave: ...
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49 views

Multiplying matrices of matrices

A question about multiplying the matrix below that is a variation Henderson's mixed model equations: |M1 M2| |M5 M6| |M3 M4| |M7 M8| whre M1-M8 are matrices. ...
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2answers
66 views

About definition of “direct sum of $p$-vector subspaces”

In the books 1 and 2, in Somme directe d'une famille de sous-espaces vectoriels, I am reading the following: 1) let $E,F$ two vector subspaces of $V$, $E+F$ is direct sum, $E+F \doteq E\oplus F$, if ...
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1answer
542 views

Finding the basis of a vector subspace of polynomials

Let V be a vector $\mathbb{R}$-space of all continuous $f(x)$ of one real variable defined on $[0,2]$. And let W denote the set of all $f(x) \in V$ s.t. the degree of $f(x) \lt 3, \: f(1)=0$. Show ...
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83 views

Infinite dimensional operator inverse

A is a linear operator on V and there exist a single operator B on V such that AB = I or BA = I. Prove that then A is monomorfic and epimorfic. On infinite dimensions, left and right inverses need ...
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28 views

Linear transformation notation: $[f]_{B_i}$

The problem states: let $B_1$ $B_2$ and $B_3$be 3 bases of $\mathbb C^3$ over $\mathbb C$ \begin{align} B_1&=\{(1,0,0);(1,1,0);(1,1,1)\}\\ B_2&=\{(1,1,0);(1,0,0);(1,1,1)\}\\ ...
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0answers
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Let $A \in Mat_{m,n}(\mathbb F)$. Suppose $A$ has pivot in $\{i_1,\ldots, i_k\}$ and $A \sim B$. Does $Col(A) = Col(B)$ ? - spanned by same columns?

Let $A \in Mat_{m,n}(\mathbb F)$. Suppose $RREF(A)$ has pivot in columns $\{i_1,\ldots, i_k\}$. Then $Col(A) = span(A_{i_1},\ldots, A_{i_k})$, where $A_{i_j}$ denote a column in $A$. However does ...
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3answers
79 views

An abelian group is finite $\iff$ the kernel of a surjective homomorphism has rank $n$

I'm doing a course on lineare algebra and I have to show the following: let $H$ be a finitely generated abelian group and $g: \mathbb{Z}^n \to H$ a surjective homomorphism. I want to show that ...
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1answer
46 views

Kernel and image of map on unit group of finite field

This question is from a rings and modules course I'm doing: Let $p>2$ be prime and $f:\mathbb{F}_{p}^{*}\rightarrow \mathbb{F}_{p}^{*}$ be given by $x \mapsto x^\frac{p-1}{2}$. Prove that $ker(f)$ ...
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1answer
31 views

Vector Q-Space of Polynomials

If V is a vector $\mathbb{Q}$-space of all polynomials with rational coefficients in one variable whose degree $\le 2$, show that $f_1 = 5, \: f_2=3+5x, \: f_3=2-x^2$ form a basis of V. So we need ...
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1answer
66 views

Solve : No. Coins = 165 | Cost Of Coins = 555 | Denominations = 2 , 5 . Find No. Of Coins Each

Linear equation in one variable: I Need To Solve This: No. Coins = 165 | Cost Of Coins = 555 | Denominations = 2 , 5 . Find No. Of Coins Each Denomination. I Tried A lot But couldn't solve it. ...
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1answer
56 views

Characteristic Polynomial on 4 by 4 matrix

I have a matrix and I need to prove that it's diagonalizable for some values of an variable or not diagonalizable at all. My thoughts are that that the easiest way to do so is by proving that the ...
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1answer
205 views

Unique decomposition of a vector space into a direct sum

Suppose I have a vector space W that is the direct product of two subspaces, U and V. So: $W=U\oplus V$ My working definition of direct product is that $W = U + V$ and $U\cap V = 0$. Now my problem ...
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99 views

Online 3D plotting tool for systems of linear equations in three unknowns

I was trying to visualize how systems of linear equations in three unknowns work but failed miserably and I started to look for an online tool which would allow me to plot all three equations but ...
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19 views

how to solve this question on linear transformation?

The straight lines $L_1 : x=0 , L_2:y=0$ and $L_3:x+y=1$ are mapped by the transformation $w=z^2$ into the curves $C_1,C_2$ and $C_3$ respectively. The angle of intersection between the curves at ...
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2answers
389 views

Rank of the Outer Product of two Vectors

I have come across the statement that the rank of the outerproduct of two vectors is always one - but why is that? Thanks
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1answer
288 views

Total unimodularity of matrix with consecutive ones property

A matrix has the consecutive ones property (often abbreviated C1P) if its every row (or column, for column-oriented C1P) is of the form $(0,\ldots,0,1,\ldots,1,0,\ldots,0)$. There is a theorem which ...
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Getting “semi” orthogonal basis from a linear independent set

Let $K_i: \mathbb{R}\mapsto \mathbb{R}^k$ are continuous functions for all $i=1,\dots,k-d$ such that for every fixed $t\in\mathbb{R}$ we have ${\cal K}_t=\{K_1(t),\dots,K_{k-d}(t)\}$ be a linear ...
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2answers
50 views

Eigenvalues and eigenvectors of an operator

I have $Ku(t)=\int_0^1 G(t,s) u(s) ds, u\in L^2[0,1]$ where $$G(t,s)=\begin{cases} s(1-t)~ 0\leq s\leq t\leq 1\\ t(1-s)~ 0\leq t\leq s\leq 1\end{cases}$$if the eingenvalues of $K$ are $1/k^2\pi^2$ ...
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1answer
47 views

Can someone explain geometric multiplicity?

I'm reading my textbook and I'm really confused about geometric multiplicity. I've read the definition and they have given an example but I'm still lost. I've tried looking it up on other websites. ...
2
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1answer
44 views

Transformations and Null spaces

Let $F$ be a field. Construct linear transforms $S,T:F^2\rightarrow F^2$ such that $S \circ T= \mathbb{0}$, but $T\circ S\neq \mathbb{0}$ Aside from the trivial case (i.e $S$ is the zero matrix), ...
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1answer
61 views

linear independent or dependent set - linear algebra

I have the following set: $\{ [1; -1; -2], [-1;0;1], [1;2;1] \}$ and I need to find out whether the set is independent or dependent. My answer and the book's answer contradict. I thought it was ...
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1answer
48 views

Find the projection matrix $P$ onto the column space of $A$

Given $A=\begin{bmatrix} 1&-6\\ 3& 6\\ 4 &8\\ 5&0 \\ 7&8\\ \end{bmatrix}$ I know that $P = A(A^{T}A)^{-1}A^{T}$. I first found $A^{T}A = \begin{bmatrix} 1 &3 &4 &5 ...