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
412 views

Best book for an engineer to refresh everything and study math for fun?

I like Calculus, Linear Algebra, Number Theory, Geometry, etc. I pretty much like everything about Math. :) The good thing about math is that all you need is a good book, a stack of paper, a pencil ...
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66 views

Anihilators question

Let $W \subset V$ is a subspace. Prove that anihilator $W^{\perp } = \{f \in V^{*}:\forall _{w\in W} f(w)=0\}$ space $W$ in $V$ is a subspace. Prove, that $W^{*} \cong V^{*}/W^{\perp} $, and if $\dim ...
2
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1answer
36 views

Inverse of specified matrix

Please help me in finding the general form of inverse matrix to matrix with $n$ rows and $n$ columns with elements except elements on diagonal, equal $1$ and elements on diagonal equal $1+x_1$, ...
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2answers
541 views

how prove GL(n,R) is not connected subset and open subset of$M_n (\mathbb{R})$with this distance

let n>1 be natural and fix number, $S:=${A : $M_n (\mathbb{R})$ be all real matrix,define this meter for all $A=[a_{ij}]$ $B=[b_{ij}]$ d(A,B):=max{|$a_{ij}-b_{ij}$|:i,j=1,2,2...,n} and GL(n,R) is ...
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Find eigenvalue via Discrete Fourier Transform

The question is as following Let $B=\begin{bmatrix} 4 & 1 & 0 & 0 & 1 \\ 1 & 4 & 1 & 0 & 0 \\ 0 & 1 & 4 & 1 & 0 \\ 0 & 0 & 1 & 4 & 1 ...
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120 views

Solve Matrix Equations

I have a system of equations I'm trying to solve: $$\begin{align}Q &= MP \\ Q^\prime &= M\Pi P\end{align}$$ $Q$, $Q^\prime$, and $P$ are all $4\times n$ matrices, and $M$ and $\Pi$ are both ...
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31 views

a question on symmetric non negative definite matrix [duplicate]

Possible Duplicate: Three linked question on non-negative definite matrices. $A$ be a real symmetric non negative definite matrix (i.e $x^TAx\ge 0\forall x\in\mathbb{R}^n$), we need to find ...
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4answers
397 views

If $X^n$ is a diagonal matrix with distinct eigenvalues, then is $X$ also a diagonal matrix with distinct eigenvalues?

Assume that there exists an invertible matrix $P$ such that $P^{-1}X^nP$ is a diagonal matrix with distinct eigenvalues, then can I say that $P^{-1}XP$ is also a diagonal matrix with distinct ...
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1answer
157 views

Show that $\ker W$ is $A$-invariant?

Consider the system.: $$\dot x = Ax$$ $$y= Cx $$ The observability matrix then will be $$ W= \left(\begin{matrix} C \\ CA \\ CA^2\\ \vdots \\ CA^{n-1} ...
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55 views

number of 1-to-1 linear functions on vectorspaces over finite fields

This is not a homework. I just ask this question myself and thought it would be easy to figure out. But I did not get the solution. Let $\mathbb{F}$ be a finite field with $|\mathbb{F}|=q$. Consider ...
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Changing variables

If there is a change of variables: $$(\vec x(t),t)\to (\vec u=\vec x+\vec a(t),\,\,\,v=t+b)$$ where $b$ is a constant. Suppose I wish to write the following expression in terms of a gradient in ...
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Is this an ODE problem, related to linear algebra? May I get a reference text?

The problem: Let $X = \{ \phi \in C^{2} [0,L] : \phi (0) = \phi (L) = 0 \}$ and we define an operator $T$ on $X$ by $$ T( \phi (x) ) = - \frac{d^{2}}{dx^{2}} \phi (x) = - \phi '' (x). $$ Then (a) ...
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5answers
135 views

Let $a,b$ and $c$ be real numbers.evaluate the following determinant: |$b^2c^2 ,bc, b+c;c^2a^2,ca,c+a;a^2b^2,ab,a+b$|

Let $a,b$ and $c$ be real numbers. Evaluate the following determinant: $$\begin{vmatrix}b^2c^2 &bc& b+c\cr c^2a^2&ca&c+a\cr a^2b^2&ab&a+b\cr\end{vmatrix}$$ after long ...
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1answer
217 views

finding the matrix and kernel of a linear transformation

Let $P_n$ denote the vector space of all polynomials in one variable with real coefficients and of degreeless than, or equal to,n, equipped with the standard basis {$1,x,x^2,…,x^n$}. Define ...
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1answer
294 views

How a rows permutation affects the SVD of a matrix?

Given a matrix with its Singular Value Decomposition: $$ X = \begin{bmatrix} x_{1,1} & \dots & x_{1,m} \\[0.3em] \dots & \dots & \dots \\[0.3em] ...
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1answer
20 views

Product of linear mapping

Matrix $A=\Bigg| \begin{matrix} 2 & 0 & 4 \\ -1 & -1 & -2 \\ \end{matrix}\Bigg|$ is a matrix of linear mapping $l: R^3\to R^2$ with the respect to bases $B = ...
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1answer
104 views

Find the condition on the real numbers $a,b$ and $c$ such that the following system of equations has a solution:

Find the condition on the real numbers $a,b$ and $c$ such that the following system of equations has a solution: $2x+y+3z=a$ $x+z=b$ $y+z=c$ After calculation I get $a-2b-c=0$ will be the answer. ...
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2answers
31 views

Computation with scalar product

Let $\vec{a}$ and $\vec{b}$ be vectors from $V_3$. Suppose, that $|\vec{a}| = 1$, $|\vec{b}|=2$ and the angle between $\vec{a},\vec{b}$ is $\frac{\pi}{3}$. Use the properties of scalar product and ...
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1answer
66 views

Errors while calculating the unknown of a matrix?

I am currently facing a problem for calculating the unknown in a matrix: The Determinant is $A=35$ and the matrix is $$A= \begin{bmatrix} 7 & 8 & 6 & u \\ -5 & 8 & 6 ...
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1answer
220 views

Let $T$ be a $4 \times 4$ matrix with real entries

Let $T$ be $4\times 4$ matrix with real entries. Suppose $T^5=0$. Then which of the following is necessarily true? (A) $T$ is the zero matrix. (B) $T$ need not be the zero matrix, but $T^2$ is the ...
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Problem related to a matrix

Taking $M$ to be of the form \begin{pmatrix} a &b &c \\ d & e & f\\ g& h & i \end{pmatrix} we get (from the $2$ given conditions) $6$ equations whereas total number of ...
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1answer
201 views

Uniqueness of Hermitian inner product

Let V be an irreducible representation of a finite group G.How to show that up to scalars,there is a unique Hermitian inner product on V preserved by G. i know of how to get an inner product. but i ...
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1answer
64 views

Find a transformation in specified basis

My task is to find a matrix of linear transformation $\varphi$ in basis $A,B$ $\varphi:\mathbb{R}^{2}\to\mathbb{R}^{4} \varphi((x_{1},x_{2}))=(3x_{1}+x_{2},x_{1}+5x_{2},-x_{1}+4x_{2},2x_{1}+x_{2})$ ...
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3answers
2k views

how to solve a system with more equations than unkowns?

In general, how do you solve a system with more equations than unknowns? I know that if I select the equations to match them with the number of unknowns, there may be zero or many solutions depending ...
2
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2answers
63 views

How to solve for these simultaneous equations

I have the following set of equations $$\pi_1 = \pi_3 + [1 - \alpha(1 - p)]\pi_4$$ $$\pi_2 = \alpha(1 - p)\pi_4$$ $$\pi_3 = \alpha(1 - p)]\pi_1$$ $$\pi_4 = [1 - \alpha(1 - p)]\pi_1 + \pi_2$$ $$\pi_1 ...
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4answers
338 views

Can a vector space have multiple spanning sets?

Maybe this is obvious, but can a vector space have multiple spanning sets or is there only a single spanning set for every vector space? Thanks
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4answers
537 views

What is an additive group?

Is an additive group a group which only has an addition operation, or can it also have other operations on it? Thanks
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2answers
158 views

How to prove that this linear operator is nilpotent?

Let $A\in M_n(\mathbb{C})$ be an arbitrary matrix , $\mathbb{C}$ is complex fields, and $L$ a mapping that is defined by $L:M_n(\mathbb{C})\to M_n(\mathbb{C})$, $L(X):=AX+XA$. How can we show that ...
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1answer
88 views

a question about functional analysis

Could you please help me to prove the following: Let $f$ be any non-zero functional on a vector space $X$ and $x_0$ is a fixed element of $X\setminus N(f)$, where $N(f)$ is the kernel of $f$. Show ...
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1answer
511 views

Three linked question on non-negative definite matrices.

1.a symmetric matrix in $\mathbb{M}_n(\mathbb{R})$ is said to be non-negative definite if $x^Tax≥0$ for all (column) vectors $x\in \mathbb{R}^n$. Which of the following statements are true? (a) If a ...
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1answer
45 views

Matrix determinant $b_{ij}=c^{i-j}a_{ij}$

I found this problem in A. Kostrikin's algebra book. There is no solution or a hint to it there. Only answer: $\det B=a$. Let $A = [a_{ij}] \in \mathcal{M}(n,n; K), \ \det A=a, \ \ c \in K, c \neq0$ ...
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0answers
101 views

Singular values of particular block matrices

Let be $\boldsymbol{S}$ an $N\times K$ real matrix having a block structure: \begin{equation} \boldsymbol{S} = \sum_{i=1}^{m} \boldsymbol{e}_i \otimes \boldsymbol{S}_i = ...
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1answer
132 views

Why a closed bounded convex set in $\mathbb{R}^{n}$ always has an extreme point?

Let $X\subseteq \mathbb{R}^{n}$ is closed, bounded convex set. How to prove that $X$ contains such point $x$ that we can't represent as $x=\frac{1}{2}x_{1}+\frac{1}{2}x_{2}$ where $x_1\in X$ and ...
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1answer
309 views

Solving a Conic Matrix given these Equations

Given a conic $\Gamma$ that has the equation $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$, $\Gamma$ can be represented by the symmetric matrix $$\mathbf{C} = \begin{bmatrix} A & B/2 & D/2\\ B/2 & ...
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73 views

The solution of the following two equations in two unknowns

I want to solve the following two equations in two unknowns: The unknowns are $x$ and $y$. Please help me.
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79 views

How does integrating the Kolmogorov forward equation give $P = \exp (Qt)$?

I originally asked this question on the stats site and I recieved an answer and after looking through the answer, I had some pure maths bits I didn't understand. The main ones were integrating ...
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2answers
86 views

Why is the inner product of polyvectors positive definite?

Let $X$ be a finite dimentional Euclidean space with the inner product $\langle...,...\rangle$, and let $k$ be an integer. Consider the polylinear form $X^k\times X^k\to{\mathbb R}$ $$ \big\langle ...
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matrix similarity upper triangular matrix

How to show: Any matrix A with real or complex entries is similar to an upper triangular matrix M whose diagonal entries are the eigenvalue of A. Thank you!
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1answer
43 views

Is there way to figure out the linear transformation by 3 given images?

I have those 3 images: T(-1,1,0,1) = (1,-1,2,1) T(1,1,1,0) = (2,3,1,-1) ...
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2answers
259 views

Condition number inequality

Let A be an invertible n x n matrix, How to show that: $K(A) \ge \dfrac{\|A\|}{\| B - A \|}$ where $K(A)$ is the condition number of the matrix $A$ and for any $B$ being an $n\times n$ singular ...
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314 views

Special Matrices which have matrix-vector multiplication complexity less than $O(n^2)$

I am looking for some special type of matrices, such that the matrix vector multiplication complexity is less than $O(n^2)$. A few such examples are Hankel and Toeplitz. But they have very less ...
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1answer
199 views

p norm Matrix relationship

I am trying to show that $\Vert A \Vert_\infty \leq \sqrt{n}\Vert A \Vert_2$ given that $A \in \mathbb{R}^{m \times n}$, $\Vert A \Vert_\infty = \max \limits_{1\leq i \leq n} \sum\limits_{j=1}^n ...
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1answer
466 views

Linearly independent subspaces

Let $E_1, \ldots, E_n$ be $n$ closed and linearly independent (but not necessarily orthogonal) subspaces of an Hilbert space $\cal{H}$, that is $(E_1 + \ldots + E_{i-1} + E_{i+1} + \ldots + E_n) \cap ...
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1answer
38 views

Calculate the thickness of material a ray passes through crossing a metal pipe

I have a metal pipe of internal radius R and wall thickness W. I fire a ray across the pipe's cross-section, perpendicular to its longitudinal axis, so that at its closest point the ray is distance d ...
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1answer
102 views

Proof of a fact about symmetric pd matrices

Several times I bumped into the following argument in my studying If $A$ is a symmetric, positive definite $n$ by $n$ matrix then there exists a nonsingular $n$ by $n$ matrix $C$ such that $A=C'C$. ...
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2answers
214 views

Formula of the determinant

Given (finite dimensional?) vector spaces $V$ and $W$, we can define the transpose of a linear map $f:V \to W$ by the obvious map $W^* \to V^*$. Can we do a similar thing for the determinant? Can we ...
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1answer
95 views

Gaussian Elimination SPD matrix

How to show that no pivoting is necessary during Gaussian elimination of a symmetric positive definite matrix? Thank you
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4answers
173 views

Let $T,S :\mathcal P \rightarrow \mathcal P$ be such that $T \circ S$ is identity

I came across the above problem and was trying to solve.Could someone point me in the right direction? Thanks everyone in advance for your time.
3
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1answer
180 views

prove that $\det T_{B} = |\det B|^{2n}$

I'm doing an exercise in Kunze Hoffman book and be stucked in this exercise about calculating the determinant of a linear operator. Can anyone help me? Suppose $H$ is the vector space of all $n ...
4
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1answer
348 views

Invariance of eigenvalues of a product of square matrices under cyclic permutation

I recently came across this proposition that the eigenvalues of a product of square matrices are invariant under cyclic permutation of the product order. Is there perhaps some group theoretic way of ...