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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12 views

Is the following Eigenvalue inequality holds or not?

Can anyone help me with the following problem? Suppose $u=(u_1,u_2,...u_n)^T$, $e=(1,1,...1)^T$, and we have $u\geq e$. Now for any symmetric matrix $A\in S^n$ with $diag(A)=0$, can we claim the ...
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26 views

The geometric multiplicity

By given this matrix: \begin{pmatrix}0&a&0\\0&0&1\\0&0&0\end{pmatrix} Why for any a which is not 0 the geometric multiplicity = 1? and why for a = 0 the g.m. = 2? I don't ...
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1answer
18 views

Matrix of a linear map Questions

Suppose n < m. Show there exists a basis $w_1...w_m$ of w for every choice of basis for v of degree n such that the last m-n rows of M(T) consist of only $0$'s for every choice of basis for w. ...
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2answers
47 views

Does adding linearly independent vectors retain linear independence?

Suppose the vectors u, v, w are linearly independent and u'=u+v, v'=v+w and w'=u+w. I'd like to check if u', v', w' are also linearly independent. I know they can be linearly independent, such as if ...
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1answer
43 views

What is a bilinear form?

I'm a CS master student and I'm reading a paper that mentions the term "bilinear form". Actually the paper mentions "bilinear regression model". But I think in order to understand what a "bilinear ...
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43 views

Linear Algebra, Eigenvalues and Eigenvectors Exercise

I have a question from an exercise. I am given a vector space over the field $\mathbb{R}^{3}$ with 2 dimensions and I am asked to find a basis of eigenvectors. I found the eigenvalues but I have ...
3
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0answers
17 views

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
38 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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2answers
28 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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31 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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0answers
14 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 ...
2
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3answers
68 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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0answers
10 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
58 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
17 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
36 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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0answers
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
65 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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0answers
45 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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0answers
18 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 ...
3
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0answers
28 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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2answers
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
103 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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0answers
42 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 ...
3
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1answer
40 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 ...
2
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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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0answers
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
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 ...
3
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2answers
59 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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0answers
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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31 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
73 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
26 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 ...
2
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1answer
53 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 ...