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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1answer
97 views

Show if $\|\cdot\|$ is a norm then $\|f(\cdot)\|$ is a norm where $f$ is linear and invertible

I want to show that if $\|\cdot\|$ is a norm then $\|f(\cdot)\|$ is a norm where $f$ is a linear and invertible function. First I need to show if $x\neq0$ then $\|f(x)\|>0$. Since $f$ is ...
1
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2answers
29 views

Finding the Matrix of a linear map $T$.

Let $T\colon \mathcal M_{22}(\Bbb R) \to \mathcal M_{22}(\Bbb R)$ be defined by: $ T\left(\begin{bmatrix} a & b\\ c & d \end{bmatrix}\right) = \begin{bmatrix} 2c & a+ c\\ b-2c & ...
1
vote
1answer
32 views

Give a bound of norm in terms of norm

Supppose $A$ is a nonsingular $n\times n$ matrix and $x,y,b,c\in \mathbb{R}^n$ satisfy $$Ax=b$$ $$A(x+y)=b+c.$$ Give a bound on $\|y\|/\|x\|$ in terms of $\|c\|/\|b\|$. Can someone give me ideas?
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1answer
83 views

Norms and invertibility of a summation

I need to show that if $X,Y$ are matrices with $X$ invertible and $$\lVert Y-X\rVert < \lVert X^{-1}\rVert^{-1}$$ then $$Y^{-1} = X^{-1} \sum_{k=0}^\infty (I - YX^{-1})^k,$$ where $I$ is the ...
3
votes
1answer
54 views

2nd smallest eigenvalues and courant-fisher

I came across the following argument in a lecture about algebraic methods in combinatorics: Suppose we have $L$, the laplacian of some graph, and $\mu_1\leq\cdots\leq\mu_n$ are its eigenvalues. ...
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votes
2answers
218 views

An example of square matrix which satisfy two conditions

Give an example of square matrix which: Have two (and no more, only two!) eigen values: $0$ and $1$. $0$ and $1$ are once eigen values. I have a big problem with this task, because I can't ...
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0answers
78 views

Structure of a fuzzy subspace

Let $V$ be a vector space over a field $F$ and let $f$ be a function from $V$ to the interval $I:=[0,1]$ satisfying the condition that for any $a \in I$ the set $V_a:=\{v \in V | f(v) \ge a\}$ is a ...
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3answers
124 views

Finding a basis for $W=\{p(x)\in \mathbb{P}_3 \mid p(-1)=p(2)=0\}.$

Find a basis for $W=\{p(x)\in \mathbb{P}_3 \mid p(-1)=p(2)=0\}.$ First off, $\mathbb{P}_n$ denotes the vector space with polynomials with degree $n$ or lower. My work/thinking: I know that for ...
1
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1answer
71 views

Sum of polynomials with no common factors

I have come across this problem in a set of exercises leading to a proof of the Jordan Normal Form. It begins with taking a polynomial $h(x)$ such that $h(L)\equiv 0$ for a linear operator $L$, and ...
1
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1answer
225 views

If two matrices commute with a nonscalar matrix resp., they commute with each other

Let $A \in M_2(\mathbb C)$ be an arbitrary matrix which is not scalar. (In what follows the matrices are elements of $M_2(\mathbb C)$.) Then let $X := \{B \mid AB = BA\}$. Here I would like to show ...
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0answers
58 views

sum of rank of matrices is equal to no of columns and rank of product of matrices

A and I are two $n \times n$ matrices. Prove that $ R(A) + R(I-A) = n + R(A - A^2 $). I used the inequality $R(A(I-A)) \geq R(A) + R(I-A) - n$ So $ R(A^2-A)) + n \geq R(A) + R(I-A)$ How to prove ...
1
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2answers
107 views

matrix representation of polynomial

Here is a polynomial $p(x,y) = (ax + by)^2$, it can be written like this $$p(x,y) = \left(\left[ \begin{array}{cc} a & b \\ \end{array} \right] \left[ \begin{array}{c} x\\ y\\ \end{array} ...
3
votes
4answers
72 views

Existence of linear mapping

I am studying for an exam in linear algebra and I am having trouble solving the following: Do linear mappings $\phi : \mathbb{R}^2 \rightarrow \mathbb{R}^2$ with the following properties exist? $1)$ ...
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0answers
49 views

Duality in Chebychev approximation

I got messed up with this problem and can't find any clue to solve this. Hope some one here can help me. Let $A$ be an $m \times n$ matrix an let $b$ be a vector in $R^{m}$. We consider the ...
0
votes
1answer
161 views

linear map vs operator: raised to power

In linear algebra, a linear map $T: V \rightarrow W$, so $T \in \mathcal{L}(V,W)$. When $T$ maps from $V$ to $V$ itself, then $T$ is an operator on $V$. Here is my problem, I read in a book saying ...
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2answers
242 views

Property of Subordinate Matrix Norm

I do not understand why the following property for Matrix subordinate norms holds: \begin{equation} \|AB\| \leq \|A\|\|B\| \end{equation} Please explain clearly as I know that it should be shown by ...
4
votes
1answer
80 views

Mark the points $\frac{3}{4}v + \frac{1}{4}w$ from $\frac{1}{2}v + \frac{1}{2}w$

Recently, I do self-learning "Linear Algebra" by using this book "Introduction to Linear Algebra, 3rd Edition" by Gilbert Strang with his lecture on MIT Opencourseware. I am having problem with one of ...
2
votes
1answer
85 views

How to solve this 0-1 linear equations?

$A=\left(\begin{matrix} 1 & 1 & 1 & 1 & 0 & 0 & 1 & 0 & 0 \\ 1 & 1 & 1 & 0 & 1 & 0 & 0 & 1 & 0 \\ 1 & 1 & 1 & 0 & 0 ...
12
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1answer
312 views

Eigenvalues of $A$ and $A + A^T$

This question has popped up at me several times in my research in differential equations and other areas: Let $A$ be a real $N \times N$ matrix. How are the eigenvalues of $A$ and $A + A^T$ related? ...
4
votes
1answer
221 views

Prove the inverse of the Hilbert matrix has integer entries [duplicate]

$1 \frac{1}{2} ... \frac{1}{n}$ $\frac{1}{2} \frac{1}{3} ... \frac{1}{n+1}$ $.$ $.$ $.$ $\frac{1}{n} \frac{1}{n+1} ... \frac{1}{2n-1}$ Does the inverse of this matrix ...
2
votes
1answer
71 views

About the existence of a certain type of linear map

I have a conjecture inspired by the following observation. If $f:\mathbb{R}\to \mathbb{R}$ is a continuous bijective function that satisfies $f(x)+f^{-1}(x)=2x$ and has a fixed point, then $f(x)=x$. ...
1
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1answer
21 views

generating a vector given other vectors in modulo 11

how to show that vector $X4=\begin{bmatrix}0 \\ 2 \\ 1 \\ 1\end{bmatrix}$ can be generated with $X1=\begin{bmatrix}9 & 1 & 0 & 0\end{bmatrix}$ $X2=\begin{bmatrix}8 & 0 & 1 & ...
1
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1answer
90 views

how to interpret theorem about polynomial factorization over modulo ring?

polynomial $X^n+a_1X^{n-1}+...+a_n \in \Bbb Z_2[X]$ doesn't have linear factors $\iff a_n(1+\sum a_i) \neq 0$. How then $f(X)=X+1$ can has no linear factors? Doesn't the condition expands to ...
2
votes
2answers
113 views

Providing a closed formula for a linear recursive sequence

I am studying for an exam in linear algebra and I have some trouble solving the following: Let $(a_n)$ be a linear recursive sequence in $\mathbb{Z}_5$ with \begin{align} a_0 = 2, a_1 = 1, a_2 = 0 ...
0
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0answers
76 views

Obstruction to construct the set of vectors in lattice

Lets consider a lattice $\mathbb{Z}^n$ with some unimodular scalar product $\mathbb{Z}^n \times \mathbb{Z}^n \mapsto \mathbb{Z}$ and the set of vectors $e_0,\ldots,e_k$ with conditions: $$ ...
2
votes
1answer
804 views

Affinely independent

I am wondering about affinely independent and just linearly independent. On Wikipedia it is explained that $u_i$ are affinely independent if $u_1 - u_0, ...,u_k -u_0$ are linearly independent. It is ...
2
votes
1answer
45 views

Gradient of $\displaystyle \left|\sum_{k=1}^Nx_k^2e^{-j\frac{2\pi}Nkl}\right|$?

How to evaluate gradient of this function ? $$\displaystyle f(\mathbf{x}) = \sum_{l=1}^{N-1} \left(|\sum_{k=1}^Nx_k^2e^{-j\frac{2\pi}Nkl}| - A\sum_{k=1}^Nx_k^2\right)^2 $$ $\mathbf{x}$ is a real ...
7
votes
1answer
250 views

Determinant game - winning strategy

I came across this problem while looking at Putnam problems a while ago: "Alan and Barbara play a game in which they take turns filling entries of an initially empty 2008 x 2008 array. Alan plays ...
0
votes
1answer
58 views

Write the normal and vector form of the equation in $\mathbb{R}^2$

This is more of a check then anything else. Here is what I have. Need to find the normal and vector form of the equation $$-2x+3y=5$$ Normal form: $$(-2,3) \cdot [(x,y) - (-1,1)]$$ Vector form: Now ...
0
votes
2answers
106 views

What is wrong with this thinking? Linear Algebra problem

Suppose $T \in L(V)$ such that each vector in $V$ is an eigenvector of $T$. Prove that $T$ is a scalar multiple of the identity. I have googled the answer with my own proof and mine looked almost ...
2
votes
2answers
135 views

Freshman's dream in the quotient

Let $k$ be a field of characteristic $p$ and let $A$ be a $k$-algebra. Let $S$ be the subspace of $A$ generated by the commutators, that is, the $k$-span of elements of the form $[a,b] = ab-ba$ (I ...
1
vote
1answer
311 views

Orthogonal projection onto an affine subspace

If we want to find the distance from a vector $x$ to a subspace $S$, we take $\| (I-P_S) x\|$, where $P_S$ is the orthogonal projection onto the subspace $S$. Obviously we could do the same thing for ...
2
votes
1answer
128 views

Prove a matrix is positive definite

Please, can somebody help me with this problem? [Ciarlet 5.3-1] Let $A$ be an invertible Hermitian matrix, with the splitting $A = M-N$, $M$ being an invertible matrix. Prove that, if the Hermitian ...
0
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2answers
144 views

ISO brief primer on special matrices

I am looking for a brief primer on the following types of matrices: stochastic, doubly stochastic, symplectic, Vandermonde, Hadamard, permutation, tridiagonal and circulant. Nothing too deep, just a ...
2
votes
4answers
634 views

Do all symmetric $n\times n$ invertible matrices have a square root matrix?

My question relates to the conditions under which the spectral decomposition of a nonnegative definite symmetric matrix can be performed. That is if $A$ is a real $n\times n$ symmetric matrix with ...
1
vote
1answer
88 views

Finding a point within a 2D triangle

I'm not sure how to approach the following problem and would love some help, thanks! I have a two-dimensional triangle ABC for which I know the cartesian coordinates of points $A$, $B$ and $C$. I am ...
4
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0answers
94 views

Is this isomorphism natural?

Suppose I constructed a linear map $\phi$ without choosing a basis, but in order to check that $\phi$ is an isomorphism, I chose a basis. Is $\phi$ still considered a natural isomorphism? Edit: The ...
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0answers
144 views

Commuting self adjoint operators

I want to prove that two commuting, self adjoint operators $A,B$ on a finite dimensional complex inner product space $V$ have identical eigenspaces. So far I have that the eigenspaces of $A$ are ...
1
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1answer
69 views

$\forall v\in V:(Tv,v)=0\implies T^{\star}=-T$

Let $V$ be a real inner product space and $T:V\to V$ a linear transformation. $$\forall v\in V, (Tv,v)=0\implies T^{\star}=-T$$ "An attempt": $$(Tv,v)=(v,T^{\ast}v)=(T^{\ast}v,v)$$
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37 views

Solving multiple linear equations

I'm a bit rusty on my linear algebra. I have the following equations: $$\operatorname{weight}_C = \frac{\frac{P_y - A_y}{B_y - A_y} - \frac{P_x - A_x}{B_x - A_x}}{ \frac{A_x-C_x}{B_x-A_x} - ...
0
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2answers
47 views

About $2\times 2$ similar matrices…

Let $A$ and $B$ be $2\times 2$ matrices with the same trace and the same determinant. Are $A$ similar to $B$? I know that they have the same characteristic polynomial. So, exist $P,Q\in GL_2(F)$ ...
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2answers
99 views

Given characteristic polynomial of $T$, need find characteristic polynomial of $T^3$

Let $T:\mathbb{R}^2\to \mathbb{R}^2 $ be a linear transformation with characteristic polynomial $x^2+2x-3$. Find the characteristic polynomial of $T^3$. How to do this? Thanks!
2
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1answer
212 views

Property for Norms of Matrices

I am having trouble with the following problem: Show that the vector norm $||x||_1$ gives the subordinate matrix norm: \begin{equation} ||A||_1 = \max_{1\leq j\leq n}\sum_{i=1}^n|a_{ij}| ...
0
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3answers
135 views

True or false: If $||Tv+v ||=||Tv||+||v||$, then $1$ is eigenvalue of $T$

Let $V$ be an inner product space over $\mathbb{C}$. And let $T:V\to V$ be an unitary transformation. Suppose that for $ 0 \neq v\in V$ we have $||Tv+v ||=||Tv||+||v||$, then $1$ is eigenvalue of ...
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1answer
39 views

Linear transformation with $T^3+I=0$

Is it true or false that if $T:V\to V$ is linear transformation such that $T^3+I=0$, then $\dim V\geq 3$?
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1answer
132 views

Linear Algebra 2: True or False question.

Let $A\in M_3(\mathbb{C})$. Suppose that $A^{\star}A=AA^{\star}$. Is the following true or not: If $(1,0,1)^{T}$ and $(1,1,0)^{T}$ are eigenvectors of $A$ with eigenvalues $\alpha,\beta$, then does ...
0
votes
1answer
35 views

Bilinear Form - Proof

I have to prove that the mapping $f(x,y) = {\displaystyle \sum_{i=1} ^ {n} }{ \displaystyle \sum_{j=1}^{n} }x_iy_j{f}(e_i,e_j)$ is a bilinear form, that is, inter alia, the condition: ...
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0answers
136 views

Matrix spectral decomposition

Let $A$ be a square matrix $(N \times N)$ and $a_{ij} \in \mathbb{R}$. Suppose A has N eigenvalues $\lambda_{1} < \lambda_{2} < ...\lambda_{n} \in \mathbb{R}$. $A$ = $R \Omega R^{-1}$ its ...
1
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2answers
69 views

bilinear form - proof

I have to prove that the mapping $f(x,y)={\displaystyle \sum_{i=1}^{n}}{\displaystyle \sum_{j=1}^{n}}x_{i}y_{j}{f}(e_{i},e_{j})$ is a bilinear form, that is, inter alia, the condition: ...
1
vote
2answers
381 views

How to calculate weight of positive and negative values.

We have used formula formula to calculate weight as, $$ w_1 = \frac{s_1}{s_1 + s_2 + s_3};$$ $$ w_2 = \frac{s_2}{s_1 + s_2 + s_3};$$ $$ w_3 = \frac{s_3}{s_1 + s_2 + s_3};$$ However, their is ...