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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Vector (scalar) product: associativity

Let $x$, $y$, $z$ be vectors of $\mathbb{R}^{n\times1}$. Consider this scalar result: $b = x^{\top} y z$. The issue is that the above product does not follow the classical associativity algerbra ...
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proving a fraction with 2 parameters to be small

Hi I have a fraction as below $$\frac{1.623x^4+0.434x^4\sum_iy_iz_i^2+(0.014x^2+0.0027)\sum_iy_iz_i^4}{1.645x^2+(0.083-0.329x^2+0.435x^4)\sum_iy_iz_i^2+0.014\sum_iy_iz_i^4}$$ where $x\in[0, 0.5]$, ...
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1answer
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System of linear algebraic equations in maple

I have to solve a system of linear algebraic equations in maple. In my book it is given using linalg package in maple, which is deprecated. So I want to use Linsolve from LinearAlgebra package, but ...
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23 views

Volume of a polytope cut off by a hyperplane

Given a maximization problem with constraints, and adding a few more constraints using the Gomory cuts and solving the relaxed maximization problem, we can arrive at integer solutions. I am looking to ...
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1answer
18 views

Rank of a simple matrix series

Problem Specifications and Given conditions I have a matrix $L$ with rank 3 and dimension $ 3 \times 3$. $L = K_0+\sum_{n=1}^{\infty}K_i $ . Rank of $K_0$ is 3 and rank of L is also 3. Rank of ...
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Matrix Rank calculation

I have a matrix A . A can be written as A=B+D. I know rank of B. It is 3. Is it possible for A to have ranks $<3$ . If so please prove.
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1answer
38 views

Rank of a Matrix Sum

I have matrices of $3\times3$ dimension such that S=A+B. I know there is one inequality connecting rank of the matrices A,B and its sum S? Could you write down that here. It will be a great help for ...
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conversion by schur or svd decomposition?

Inorder to perform eigen decomposition, I converted a rectangular matrix to square by multiplying with the transpose of the matrix. After decomposition, I got the component matrices. If I multiply the ...
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1answer
71 views

Project sin(x) onto orthonormal basis that span ${(1, x, x^2, x^3, x^4, x^5)}$ on domain $[-\pi, \pi]$

I am self-studying LA through Linear Algebra Done Right 2nd ed. I probably made a blatant error somewhere but I have been stuck for a whole day now. The book gave the answer $0.987862x − 0.155271x^3 ...
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Matrix Inversion Test ( Sum of Matrix series)

Friends,I have a set of matrices of dimension $3\times3$ called $A_i$. , Following are the given conditions a) each $A_i$ is non invertible except $A_0$ because their determinant is zero. b) ...
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Accuracy of line intersecting algorithem decrase with large precisions

from the above pic I found the value of x from equation of line p1-p2 and perpendicular line from point a to the Line(p1,p2) .The intersecting point is X ,but the accuracy is less see the result ...
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+500

vector spaces whose algebra of endomorphisms is generated by its idempotents

Let $V$ be a $K$-vector space whose algebra of endomorphisms is generated (as a $K$-algebra) by its idempotents. Is $V$ necessarily finite dimensional? EDIT (Jul 26 '14) A closely related question: ...
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How to prove $\sum_{i=1}^k(\frac{1}{\alpha_i}\prod_{j\neq i}^k\frac{\alpha_j}{\alpha_j-\alpha_i})=\sum_{i=1}^k\frac{1}{\alpha_i}$?

How to prove $\sum_{i=1}^k(\frac{1}{\alpha_i}\prod_{j\neq i}^k\frac{\alpha_j}{\alpha_j-\alpha_i})=\sum_{i=1}^k\frac{1}{\alpha_i}$? Where $\alpha_1, \alpha_2,\ldots, \alpha_k$ are $k$ distinct ...
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106 views

How to find exponential of triangular matrix

I'm studying for an exam and I can't find this in my notes or in the book, but it's on a past exam... Given $A = \begin{bmatrix}-1 & 1\\0 & -1\end{bmatrix}$, $e^{tA} = \begin{bmatrix}e^{-t} ...
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What do you call the following operations on a symmetric matrix?

Suppose we have a symmetric matrix of the following form, where the diagonal is always zero: \begin{array}{cccc} 0 & 1 & 1 & 0\\ 1 & 0 & 1 & 1\\ 1 & 1 & 0 & 0\\ 0 ...
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Is the matrix form of the cross product related to bilinear forms.

The cross product of two vectors $\mathbf{x}, \mathbf{y} \in \mathbb{R}^3$ can be represented as a matrix product as follows, if $\mathbf{x} = (x_1, x_2, x_3)^{\top}$ then $\mathbf{x} \times ...
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Linear Transformation from V to W (bijective) Show that T(v) is a basis of W if B is a basis of V.

$V, W$ two vector spaces and $T: V \to W$ is a bijective linear transformation. $B$ is a basis of $V$. Prove that $\{T(\mathbf{v}) | \mathbf{v} \in B\}$ is a basis of $W$. I started by doing ...
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22 views

Can I convert between a rotation about an axis and a rotation according to two angles (all in 3D) without solving a system of nonlinear equations?

I am writing a program that needs to be able to switch between a rotation described by 2 angles to a rotation described an axis and one angle. I found one way to do this from this question, which ...
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37 views

How to further simplify this equation?

Given that V is an invertible $n$x$n$ matrix and $\Sigma$ is a diagonal rectangular $m$x$n$ matrix, U is an $m$x$m$ matrix, b is an $m$x1 matrix and $\lambda$ is a positive number, how do u further ...
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1answer
46 views

Find the equation of the linear transformation of an orthogonal projection on the line y=mx.

Let $T : \mathbb R^2 → \mathbb R^2$ the orthogonal projection on the line $y = mx$. Prove that for all $a, b \in \mathbb R$, $$\begin{align}T((a,b)) = {\frac{1}{m^2 + 1}}(a+mb, ma + ...
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Given a satisfactory real number = [any integer]/(2b) where a and b are integers, how would one find the minimum value of b?

For instance, 0.625 = 5/(2*4). Given 0.625, how would one find 4? 0.75 = 1/(2*2). Given 0.75, how would one find 2? I should ...
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1answer
35 views

Keeping the arc length constant between points in a spiral

I'm making a visualization of points in a logarithmic spiral and want to keep the arc length between points (image particles) constant. I read that in an Archemedian spiral arc length is ...
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2answers
62 views

Is there a way to determine the matrix of $\Lambda^k(T)$ given the matrix of $T$?

Let $T$ be an endomorphism of a finite dimensional vector space $V$. Suppose that $(v_1,\ldots v_n)$ is an ordered basis of $V$. And let $[T]$ be the matrix of $T$ with respect to this basis. Is ...
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3answers
64 views

How to solve this Quadratic Word problem?

This is the word problem. If they work together, John and Vince can finish their project in Biology in two days. If they worked individually, it will take John three days longer than Vince to ...
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46 views

Linear Algebra, Quadric form

Question from an exercise Let $V$ be a vector space over a field $F$ with $charF\neq2$. If $\varphi,\psi\in V^{\vee}$ are linear functionals, we will define $\varphi\cdot\psi \colon V \rightarrow ...
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70 views

relative sign in Hodge star of tensor product

Let $V$ be a vector space of arbitrary (finite) dimension and let $(V, \langle \ ,\ \rangle, I) = (W_1, \langle\ ,\ \rangle_1, I_1) \oplus (W_2, \langle\ ,\ \rangle_2, I_2)$ be a direct sum ...
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1answer
134 views

Trace of symmetric positive semidefinite matrix when diagonalized (as a bilinear form) in a non-orthogonal basis

Let $\mathbf{S}$ be symmetric positive semidefinite matrix (i.e. one with all eigenvalues real and non-negative). Then there is an orthogonal matrix $\mathbf{U}$ (with its columns forming an ...
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1answer
24 views

Integer QR decomposition

Let $A$ be a real $m\times n$ matrix and $A=QR$ be the QR decomposition of $A$. For what integer elements of $A$ do $Q$ and $R$ have integer elements? I think there are two approaches: Constructing ...
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33 views

What is the minimum number of sign patterns in $\frac n2$ of columns (or rows) of Hadamard matrices?

Given a Hadamard matrix of size $n$, I want to know what is the minimum number of unique sign patterns in any $\frac n2$ columns (or rows). I count a sign pattern and its negation to be the same. My ...
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Invertible Linear Maps Proof [closed]

1) Suppose $V$ is finite dimensional and $S$, $T$, $U \in L(V)$ and $STU = I$. Show $T$ is invertible and $T^{-1} = US$. 2) Suppose $V$ is finite dimensional and $R$, $S$, $T \in L(V)$ are such that ...
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Linear Operators Injectivity and Surjectivity

Suppose T $\in L(P(R))$ is such that T is injective and deg Tp $\leq$ deg p for every nonzero polynomial p $\in P(R)$. Prove that T is surjective and that deg Tp = deg p for every nonzero p $\in ...
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1answer
22 views

matrix function onto and 1-1

I have just started a linear algebra paper and we are doing 1-1 and onto functions. I understand in theory what they mean, I just don't know how to prove them. For example: Define $f: ...
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1answer
51 views

Counterexample of $\text{Null}(T)^{\bot} = \text{Im}(T^{*})$

I know that $\text{Null}(T)^{\bot} = \text{Im}(T^{*})$, where $T^{*}$ means the adjoint operator of a linear operator $T$, holds when the domain of $T$ is finite-dimensional. However, the proof uses ...
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Invertibility of $I-AB$ [duplicate]

I got a question in linear algebra: 1) Let A and B be $n\times n$ matrices. If $I - AB$ is an invertible matrix, then prove that $I - BA$ is invertible. Can someone tell me how to solve this ...
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1answer
31 views

How to write this system in the form Ax=b

Given the following system of N equations with N unknowns, with $\lambda$ known and the $a_{ij}$'s also known entries of an m*n matrix A. How would you express the system in the form A x=b? x is of ...
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If $A^2$ is the zero matrix, show that $A$ is linearly dependent?

The original question was show that $0$ is an eigenvalue for the matrix $A$. This was a straightforward practice of righthand multiplication of $Ax = \lambda x \Rightarrow AAx = A \lambda x ...
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Nullity of a matrix

How would I do D3 and D4? I am completely lost
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Proof of Strong Duality via Farkas Lemma

I am trying to prove what is often titled the strong duality theorem. There is a hint in the book that I'm following, and I want to follow the method they have outlined for me. I will outline the ...
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3answers
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Strange proof of Schwarz Inequality with Pythagorean Theorem

Does anyone know what is going on in this proof of the Schwarz inequality? Most importantly: how can one assume that $c^2\leqq \|A\|^2$, or later on, that $c^2\|B\| \leqq \|A\|^2$? This would imply ...
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Why does the Fourier transform include the base of the natural logarithm, the square root of -1 and $\pi$?

The formula itself, as a vector of summations of products of the original coefficients with some weight, itself a function the original and transformed coefficient indices, is not a hard pattern to ...
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Show that the image of a linear transformation is equal to the kernel

Let $\phi$ be a linear transformation such that $\phi: V\to V$ We are given the following facts: $\dim(V) = 8$ $\dim(\mathrm{Im}(\phi)) = 4$ $\phi\circ\phi=0$ Show that $\mathrm{Im}(\phi) = \ker ...
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Prove that if the sum of each row of A equals s, then s is an eigenvalue of A. [duplicate]

Consider an $n \times n$ matrix $A$ with the property that the row sums all equal the same number $s$. Show that $s$ is an eigenvalue of $A$. [Hint: Find an eigenvector] My attempt: By definition: ...
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Is a given vector in Range of a matrix?

How do I show that a given vector is in range of a matrix without solving for variables. Is there are way of doing this with null space?
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prove the following $(A^t)^{-1}=(A^{-1})^t$

$(A^t)^{-1}=(A^{-1})^t$ Proof: $(A^{-1})^{t}*A^T=(A*A^{-1})^t=I$ How to continue from here?
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Logarithmic spiral appears inverted

I'm learning to code the equation for a logarithmic spiral for a graphics visualization. However, it appears to be inverted with the radius getting smaller (rather than larger) toward the outside of ...
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2answers
41 views

Subspace Equations

I'm studying the book "Finite Dimensional Vector Spaces" by Paul Halmos. I'm doing q5 from $\S 12$ Dimension of a Subspace, in chapter $1$. I'm not all that used to L.A. proofs, so I'm looking for ...
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Prove that there is a basis of a lattice $\Lambda$ s.t. a reflection is of a certain form

Consider $\Lambda$ a lattice in $\mathbb{R}^2$. Let $S \in O(\Lambda)$ be a reflection, i.e. $\det S = -1$. Set $S_{1}= \begin{pmatrix} 1 & 0 \\ 0 & -1 \\ \end{pmatrix}$ and $S_{2}= ...
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Iterative methods for solving a linear equation system

There are several methods known for solving a linear equation system Ax = b (like Jacobi or Gauss-Seidel) by iterating $x_{n+1}=Mx_n+c$ with a matrix M, for which some norm is smaller than 1. But ...
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Question about proving symmetric matrices are diagonalizable

Definition : If a n by n matrix $A$ is orthogonally congruent to another matrix $B$, then there exist an orthogonal matrix $C$ such that $$A = C^{-1}BC$$ Theorem: If $A$ is symmetric, then $A$ is ...
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$\ker(A)=\text{Im}(A^*)^\perp$

How do I show that $\ker(A)=\text{Im}(A^*)^\perp$ for any square matrix $A$. I have done this problem before with the linear operator $T$ on a hermitian space but I can't seem to apply what I have ...