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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“The Conjugate of a matrix”

I am having some trouble understanding a definition/question in my linear algebra text book. The question states " If $A$ is a square matrix, a matrix of the form $P^{-1}AP$ where $P$ is invertible ...
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When does Ax +b = 0 not have a solution? [duplicate]

And does that tell us anything about the properties of matrix A?
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Alternative introduction to tensor products of vector spaces

One of the main obstacles in understanding the tensor product is that, unlike many other algebraic structures, you cannot really get hold of its element structure. This confuses many beginners. The ...
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Use eig and svd syntax in matlab to find complex eigenvalues of a matrix

For matrix $A= \left( \begin{array}{c} 1 & 1 \\ -1 & 1 \\ \end{array} \right) $ when I calculate the eigenvalues (without matlab) , I find $\lambda_1=1+1i$ and $\lambda_1=1-1i$ and when I ...
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Find all possible values of rank(A) as a varies?

$$ A= \begin{bmatrix} a & 2 & -1 \\ 3 & 3 & -2 \\ -2 & -1 & a \\ \end{bmatrix} $$ I started to reduce the matrix but got stuck. Is ...
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Specific vector space isomorphism preserving a special range.

Let $V = C^0 (\mathbb{R}$) be the vector space of continuous functions $f : \mathbb{R} → \mathbb{R} $ (over $\mathbb{R}$). $U1 := \{f ∈ V : f(x) = f(x + 1) \forall x ∈ R\}$ $U2 := \{f ∈ V : f(x) = ...
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Proving from (a/b + b/c + c/a) being natural that abc is some natural number cubed [duplicate]

$a, b, c \in \Bbb Z$ $a, b, c, \gt 0$ $(\frac {a}{b} + \frac {b}{c} +\frac {c}{a}) \in \Bbb Z$ How to prove that $abc$ is an integer cubed?
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How to I see that $n-1$ linearly independent constraints $a_i^Tx \ge b, i \in \{1,\dots, n-1\}$ define a line in $\mathbb R^n$?

How to I see that $n-1$ linearly independent constraints $a_i^Tx \ge b, i \in \{1,\dots, n-1\}$ define a line in $\mathbb R^n$ ? ($b, a_i, x \in \mathbb R^n$) If I can find a vector $p$ that ...
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Determine whether $b$ is in col$(A)$ and whether $w$ is in row$(A$)?

Okay, so I have given this problem A shot. I got the answer so col$(A)$ however, I was just wondering if I did it correctly? Here is my work: $$ A= \begin{bmatrix} 1 & 1 & -3 ...
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true or false about linear tranformation

Let $T: V \rightarrow V$ an operator $Tx_i = a_ix_i, i = 1, 2, a_1 \neq a_2$. Prove or give counter example 1. $\{x_1, x_2\}$ is a linearly independent set. 2. Let $y$ eigenvector of $T$ with ...
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If $A$ is a $2\times 2$ matrix with a repeated eigenvalue $r$, then $\mathrm{e}^{At}=\mathrm{e}^{rt}\left[I+(A-rI)t\right]$

If $A$ is a $2\times 2$ matrix with a repeated eigenvalue $r$, show that $\mathrm{e}^{At}=\mathrm{e}^{rt}\left[I+(A-rI)t\right]$. I have already been able to show that if $A$ is a general ...
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1answer
36 views

Factorizing Cubic Equations.

Factorization of Cubic Equations has always obstructed my way to the solution to a problem. Is there any simple technique to factorize them?
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28 views

Show that the following four conditions of a linear map are equivalent

Let $\theta:V \rightarrow W$ be a linear map, where $V$ and $W$ are finite-dimensional. Prove the following four conditions are equivalent: $\theta$ is injective (or one-to-one) $\theta$ is onto ...
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R3 to Planar Subspace Tranform

I'll ask my question three ways to try to maximize my chances of successful communication. I have: a point 'P' in R3 with coordinates $(P_X,P_Y,P_Z)$ A plane Defined by: A point 'O' given by ...
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Basis span of space necessary to be orthogonal?

Q1 If a vector space V that span of {v1,v2,....,vk},can the basis vector of V are not mutually orthogonal? (From several users comments, the answer is the basis vector can be no mutually ...
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Finding a basis for a subspace in $\;\Bbb R^4\;$

I know this might be a really simple question to ask but I just don't understand how to obtain the answer to this question. I've tried to understand subspaces (and even the difference between a space ...
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matrx with orthonormal columns and multiplication with its transpose

Let $U$ mxn matrix with orthonormal columns. So, we have $U^TU=I$ s.t. $I_{nxn}$ identity matrix. I want to see whether $UU^T=I_{mxm}$ holds. Is my attempt correct? Attempt: suppose $UU^T \neq ...
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60 views

why algebraic numbers form a field? [duplicate]

of course the problem is how to prove if a and b are both algebraic real numbers then ...
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1answer
36 views

A question on the Wronskian

Let $f(z),g(z)$ be two complex-valued functions defined in some domain $D$. Suppose we want to show that $$f(z)+g(z)\neq 0 \tag1$$ for all $z\in D$. I think I'm right in saying we can use the ...
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how to solve liner equations with decimal values like 2n=0.58(12 - n)

I am trying to solve problems given in the book "Algebra - A Complete Introduction". The author in chapter Liner Algebra has given some examples how to solve a liner equation which I understood well. ...
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Find a matrix C in Reduced Echelon Form that is row equivalent to A [on hold]

If A is |1 -2 0 2| |2 -3 -1 5| |1 3 2 5| |1 1 0 2| Find a matrix C in Reduced Echelon Form that is row equivalent to A
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Proof that the span of a list is equal to the span of any reordering of the list

Claim: If $(w_1,w_2,...w_m)$ is an arbitrary reordering of $(v_1,v_2,...v_m)$, then $span(w_1,w_2,...w_m) = span(v_1,v_2,...v_m)$. Proof By definition, ...
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Write the equation in matrix form and also find the value of h if the coefficient matrix is singular [on hold]

Write the equation $2x+hy=1$ and $3x+2y=2$ in matrix form and also find the value of $h$ if the coefficient of the matrix is singular.
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$\mathbb{u,v,w}$ are vectors in 3 dimensional space-

$\mathbb{u,v,w}$ are vectors in 3 dimensional space. Under what restrictions on $c,d,e,$ will the combinations $c\mathbb{u} + d\mathbb{u} + e\mathbb{v}$ fill in the triangle formed by the heads of the ...
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Extracting the dual feasible search directions for the primal-dual potential reduction algorithm?

I am trying to implement the 4.4 Primal-dual potential reduction algorithm introduced in M.S Lobo et al.. Here is a screenshot depicts the algorithm flow: As ...
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A question about similarity transformation.

Say $A$ is an $n\times n$ symmetric matrix such that every row (and hence column) has exactly $d<n$ non-zero entries. Does there exist similarity transformations on $A$ which will maintain these ...
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gram matrix determines vectors up to isometry

According to http://mathworld.wolfram.com/GramMatrix.html, the gram matrix determines a set of vectors up to an isometry. I'm trying to prove this statement. More specifically, let $A, B \in ...
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37 views

orthogonal projection onto linear space of matrices

Let $M$ be an $n_1 \times n_2$ matrix with rank $r$ and let $M = U\Sigma V^T$ be its SVD. Define the space $T = \mathrm{span}\{\{ u_k y^T : y \in \mathbb{R}^{n_2}, 1 \leq k \leq r\} \cup \{ x v_k^T : ...
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How can I define T in this linear transformation?

For $T: V2\to V2$ $T$ maps each point with polar coordinate $(r,\theta)$ to each point with polar coordinate $(r,2\theta)$ and $T$ maps $0$ onto itself. I let $r= \sqrt{ x^2 + y^2}$ and ...
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1answer
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Is there a way to do this besides brute force?

$A$ is a $d\times n$ matrix and $\mu>0$. I'm trying to show that $$(AA^T + \mu I)^{-1} A = A(A^T A+\mu I)^{-1}.$$ The only way I've thought about doing this was by the brute force method of ...
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3answers
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General nilpotent matrix to upper right matrix

Is it possible to explicitly give the basis transform matrix $Y$ for transforming a nilpotent 2-by-2 matrix $A$ to a matrix, whose only nonzero entry is in the upper right corner? $Y^{-1}AY=\left( ...
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Intro Linear Algebra Proofs

Hello I am having some trouble coming up with a solution to some text book problems. "If A is a an invertible n x n matrix, show that AX=B has a unique solution for any n x k matrix B." Im not sure ...
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Find system of equation with infinite solutons?

$$2x+7y-5z=0\\ 5x-2y+6z=1\\ 7x+5y+z=1$$ Answer should be in the form of (blank,blank,z) where z is any real number.
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Has the degree to which a partial eigensystem of a large sparse matrix approximates the complete eigensystem been determined?

Does anyone know of any studies or results regarding the degree of approximation or the error in estimating the complete spectrum of a large sparse matrix by means of its first $n$ eigenvalues and ...
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What are these formulas that are suppose to be Gram-Schmidt

The formulas are in this picture. My question is what are these formulas used for. I tried using them but they don't work. I'm familiar with Gram-Schmidt but these don't look like GRAM. I got these ...
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reflection(reflection) = rotation

Lel $\alpha$ and $\beta$ be two distinct simple roots in a root system $\Phi$. How to prove that i) $S_{\alpha} S_{\beta}$ is a rotation in $\mathbb{R}\Phi$ ii) Composition of two reflection is a ...
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Linear Algebra - args complex number question

I need to solve this problem : $$z^3-(2+2i)^2=0$$ This is what I did : $$z^3 = (2+2i)^2$$ $$z^3 = 8i$$ The formula for args is : $$\tan(args)=\frac{b}{a}$$ in this case its clear that the args is ...
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Condition that multiplied hermitian matrix stays hermitian

Suppose we are given a hermitian matrix $E \in \mathbb{C}^{n\times n}$. I want to find sufficient conditions on the entries of a real symmetric matrix $M$ (depending on the entries of the given ...
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2answers
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Would Evaluating a polynomial at uniformly random points outputs random values?

I`m wondering if we evaluate a polynomial on many points picked uniformly at random. Can we say the output values Y's are uniformly at random?
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A matrix equation with real coefficients

The problem is the following: Find $\lambda$ such that $ b^{T}A\left[A^{T}A-\lambda L^{T}L\right]^{-1}L^{T}L\left[A^{T}A-\lambda L^{T}L\right]^{-1}A^{T}b-\delta^{2}<0 $ where ...
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Positive matrices are open

An linear application $A:\mathbb R^n\to \mathbb R^n$ is positive when it is symmetric and besides that $\langle Ax,x\rangle\gt 0$ for every $x\neq 0$ in $\mathbb R^n$. I would like to prove the set of ...
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Novel approaches to linear algebra and geometry

I'll be studying Brannan's Geometry and Lang's Introduction to Linear Algebra for one university course. I would like to know if you can you suggest some books that offer a unique perspective on the ...
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show that dim(L,W) = mn

There are two finitely dimension vector spaces $V$ and $W$. Dimensions are $n$ and $m$ respectively. $$L(V,W)=\{T:V\rightarrow W \;|\; T \;\text{is linear}\}$$ $L(V,W)$ is a vector space with ...
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Growth rate, annualized growth

At the end of 2001 we had 1,000,000 oranges. 55% of those oranges were stale. What would be the annualized growth rate needed to increase non-stale by 75,000 oranges over four years? Show your ...
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3answers
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Why solving a system of linear equation produces the intersection of the equation

1) $x+y=1$ 2) $-x+y=1$ Geometrically we can visualize the two lines will intersect at $x=0, y=1$. Consider this algebraic solution using Gaussian Elimination, . But why do they be reduced to the ...
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1answer
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Closed form solution for a 3x3 matrix given some constraints

I would like to know if it's possible to find a closed form solution (even if not unique) for the $3\times3$ rank-deficient matrix M meeting the following constraints (in the equations below, $x,y$ ...
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To prove that the sum of the roots of the characteristic polynomial of a square matrix is equal to the trace of the matrix

How do we prove that the sum of the roots of the characteristic polynomial of a square matrix is equal to the trace of the matrix ? I want a proof which does not use much computation or determinants ; ...
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What are the x-intercept for the graph below? [on hold]

What are the x-intercept for the graph below?
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29 views

Minimize the Frobenius norm of the difference of two matrices with respect to matrix: $\underset{B} {\mathrm{argmin}} \left\| A- B \right\|_F$

The following question is similar to this one, but I think that it is not straightforward to move from one to the other, so please take a look. Otherwise, please let me know and I will delete it. ...
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1answer
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Find point on a line using its orthogonal projection

How to find a point $\bf{X}$ on a line from its orthogonal projection $\bf{P}$ on another line. Lets say we have vectors $\bf{A}$, $\bf{P}$ which start at $\textbf{0}$, how to find point $\bf{X}$? ...