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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Geometric meaning of a matrix decomposed into its symmetric and skew-symmetric parts

What's the geometric meaning of a matrix decomposed into its symmetric and skew-symmetric parts? For example, a skew-symmetric matrix on its own can be interpreted as an infinitesimal rotation. As ...
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15 views

Show that a matrix with monomial entries is invertible [duplicate]

Let $z_1,z_2,...,z_{n+1}$ be distinct non-zero real numbers. Show that ...
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1answer
34 views

Can I always extend an inner product from a real to a complex vector space?

Let V be a vector space over the real numbers with finite dimension. Let <,> : VxV -> R be an inner product on V. Let W be the same vector space V, but now considered as a vector space over the ...
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22 views

Proving boundedness of a function.

Consider the function \begin{eqnarray} f(x_1,x_2,\cdots, x_n) = \frac{\sum_{i}^{n}a_ix_i}{\sum_{i}^{n}b_ix_i}, \end{eqnarray} over the set $S = \{x := (x_1,x_2,\cdots, x_n):-1 \leq x_i \leq 1,\; ...
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29 views

What is the difference between the span of a set to its subspace?

I am confused with some of the definitions of linear algebra. I know that the span of set S is basically the set of all the linear combinations of the vectors in S. The subspace of the set S is the ...
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1answer
24 views

Finding equation of line with given slope

Find the distances of the point (1,2) from a straight line. The slope is given to be 5 and the line passes through the intersection point of the lines $x+2y = 5$ and $x - 3y = 7$ Obviously I could ...
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13 views

Set and List distinction - Golan Linear Algebra book

He says: A finite or countably-infinite selection of elements of a set A is a list. then he says: Note that the elements of a list need not be distinct: Taking the case of a list with ...
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2answers
51 views

If A+tB is nilpotent for n+1 distinct values of t, then A and B are nilpotent.

Suppose A and B are $n\times n$ matrices over $\mathbb{R}$ such that for n+1 distinct $t \in \mathbb{R}$, the matrix A+tB is nilpotent. Prove that A and B are nilpotent. What I've tried so far: ...
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1answer
27 views

Can unitary matrices be defined for infinite-dimensional space?

As the title says, can unitary matrices be defined for infinite-dimensional space?
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9 views

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
18 views

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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1answer
56 views

solving the equation

let there be a function $ f(x)= \ln x-kx^2, k>0$ determine for whihc values of $ k$ ,the equation $f(x)=0.5$ has a single solution; attemp to solve: $$0.5 = \ln x-kx^2$$ $$kx^2 +0.5 = \ln x $$ ...
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1answer
19 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
16 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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1answer
12 views

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
35 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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7 views

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
62 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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44 views

How to prove the following rank problems [on hold]

I am quite confused with this question. How can I initiate and approach a solution. Thanks!
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2answers
44 views

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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17 views

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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27 views

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?
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53 views

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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2answers
102 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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40 views

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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1answer
39 views

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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1answer
28 views

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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1answer
21 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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31 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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4answers
35 views

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
34 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
58 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
59 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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32 views

Linear Algebra, Quadric form

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

relative sign in hodge * of tensor product

For $\delta_i \in \bigwedge^{k_i}W_i^*$, $i=1,2$, the Hodge $*$-operator of $\delta_1 \otimes \delta_2$ is given by $$ *(\delta_1 \otimes \delta_2)=(-1)^{k_1k_2}(*_1\delta_1) \otimes(*_2 \delta_2)$$ ...
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1answer
70 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
22 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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1answer
28 views

What is the 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 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 guess ...
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2answers
24 views

Invertible Linear Maps Proof [on hold]

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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1answer
11 views

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
20 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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1answer
52 views

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
27 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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5answers
77 views

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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2answers
37 views

Nullity of a matrix

How would I do D3 and D4? I am completely lost
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24 views

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
98 views

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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48 views

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 ...