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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Linear Independence with distinct variables

If there is a group of vectors $v$ such that $v=\left(\begin{array}{c} 1\\1 \end{array}\right), \left(\begin{array}{c} x_1\\x_2 \end{array}\right), \left(\begin{array}{c} x_1^2\\x_2^2 ...
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Case of affine hull and linear hull possibly being euqal

Let C be a set in $\mathbb{R}^n$. Let $aff(X)$ denote the affine hull of $X$, and $lin(X)$ denote the linear hull of $X$. Suppose $x \in aff(C)$. Then, is it true that $aff(C-x)=lin(C-x)$? The ...
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Is this Determinant and Trace identity equivalent to Unitary matrix?

Thanks for any help in advance. I have this equality for a 2x2 invertible complex matrix: $$\text{Tr}(AA^*)=2|\text{det}(A)|^2$$ where $*$ is complex conjugate transposition. Is this equality ...
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16 views

Jacobi vs. Gauss-Seidel: convergence

I know that for tridiagonal matrices the two iterative methods for linear system solving, the Gauss-Seidel method and the Jacobi one, either both converge or neither converges, and the Gauss-Seidel ...
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64 views
+50

When is there a vector $D$ with positive coordinates such that $e^{Ct}D$ has a negative coordinate?

Let $C$ be a $2 \times 2$ asymmetric matrix with real entries. Assume that $C$ has strictly negative, real eigenvalues. Fix $D\in\mathbb{R}^2$, where $D > 0$ (i.e., both coordinates are strictly ...
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Least squares fitting using cosine function?

Hello I am trying to fit a harmonic of the form $$y = b + c\cos(x)$$ to four data points (0,6.1) (.5,5.4) (1,3.9) (1.5,1.6) using least squares for homework. I know that the error $= Y_i - f(x_i)$ but ...
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0answers
349 views

Frobenius Inequality Rank

I was looking for an answer for this problem in terms of matrices, but I really don't know how to prove this result. The proposition says that: Let $A\in M_{m\times k}(\mathbb{C})$, $B\in M_{k\times ...
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Positive Semi Definite Matrix

If $A$ is a positive semi definite matrix, is $\left[ \begin{matrix}c_1A & c_2A \\ c_3A & c_4A\end{matrix} \right]$ positive semi definite? ($c_1, c_2, c_3, c_4 > 0)$ In general, what ...
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Prove that if $S,T:E\longrightarrow F$ are linear transformations, then $\big|r(T)-r(S)\big|\le r(T+S)\le r(T)+r(S)$, where $r(T)=\dim Im(T)$.

I would like to know if this proof is right. In any case, anyone may feel free to provide a solution to the given problem. Prove that if $S,T:E\longrightarrow F$ are linear transformations, then ...
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a question about definition m-tuples [on hold]

Let $m>0$ and we have an $m$-tuple of natural numbers. For example we know that $\varepsilon_2=(0,1,0,\ldots,0)$. Now question is that : Does it make sense for $\varepsilon_0$ ?
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$T:E\longrightarrow F $ isomorphism. Prove $\dim \langle \vec x_1,\ldots ,\vec x_n\rangle =\dim \langle T(\vec x_1), \ldots,T( \vec x_n)\rangle$.

Let $T:E\longrightarrow F $ be an isomorphism. Prove $\dim \big\langle \vec x_1,\ldots ,\vec x_n\big\rangle =\dim \big\langle T(\vec x_1), \ldots,T( \vec x_n)\big\rangle$, for every $\{\vec x_1, ...
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Polar correlation and conics in RP2

I'm stuck on a small detail in Proposition 1.2.8 in Geiges' Introduction to Contact Topology. Let $C$ be a conic in $\mathbb{R}P^2$ given by $q^tAq=0$, where $A$ is a nonsingular, symmetric 3x3 ...
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44 views

Vector space without a scalar product

In linear algebra the terms vector space and scalar product always (at least for me) appear together. Can you give me an example of a vector space without a scalar product? Does the senescence Let V ...
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46 views

$\vec{a} \times \vec{b} = \vec{c} \times \vec{d}$ . what can you say about the direction of $\vec{b} \times \vec{c}$?

I know that $\vec{a} \times \vec{b}$ and $\vec{c} \times \vec{d}$ are perpendicular therefore the dot product would equal $0$.
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Proof of Laplace expansion using minors

I've come across with the following proof of the Laplace expansion: Let $\Delta=\sum_{j=1}^n (-1)^{1+j} a_{1j}\bar M_j^1$ and $\tilde{\Delta}= \sum_{j=1}^n (-1)^{i+j} a_{ij}\bar ...
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1answer
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Questions on proof that for $U$-invariant diagonalisable maps, the restriction $A_{|U}$ is diagonalisable too

A linear map $A \in \mbox{hom}(V,V)$ is called diagonalisable iff $$ V = \oplus_{i=1}^m \mbox{Eig}(a_i) $$ i.e. $V$ is a direct sum of eigenspaces of $A$. I have some questions on the following ...
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374 views

Ask for alternative solution

Given $n\times n$ real matrices $A,B,C,D$ such that: $AB^T$ and $CD^T$ are symmetric $AD^T-BC^T=I$ Prove that $A^TD-C^TB=I$ The solution I have come up with after a very long time is to consider: ...
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19 views

Linear system and subspaces

Let $S $ be a subspace of $R^n$ with dimension k and $m = n-k.$ Show that $$\exists A \in R^{m\times n}, b\in R^m$$ Such that $$S = \{ x \in R^n : Ax = b\}$$ My attempt consist of getting m ...
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1answer
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Linear spam of union [on hold]

Let $A = \{(0,2,-1,0,1),(0,0,3,-1,2), (0,4,-5,1,0) \}$, let $S = [A]$ and let $v = (0,m,-m,1,1) $. Determine all $m$ that make $v \in S$ true. If $w \not \in S$, is $[A \cup \{w\}] = \{(x,y,z,s,t) ...
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How prove this matrix inequality $\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)$?

Question: let matrices $A,B,C\in M_{n}(C)$ be Hermitian and Positive definite matrices, such that:$$A+B+C=I_{n}$$ Show that: $$\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)$$ ...
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20 views

Find base and dimension of given subspace

Let $T$ $\in M_{4}(\mathbb R)$ and consider $S= \{M \in M_{4\times1}|T.M = 0\}$. In the case ...
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21 views

Span and dimension of a basis [on hold]

I have to prove that there does not exist a generating set for $\bf x$ with less of $n$ vectors when $n$ is the dimension of the basis of $\bf x$. Help please!
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29 views

Statistical independence to linear independence

Suppose I have $N$ continuous independent random variables (random vectors) defined on $\mathbb{R}^N$. Can I comment on the probability of a particular realization of these N vectors being LINEARLY ...
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15 views

How to know which variables to parametize in a large matrix?

(dont want anyone to solve the problem, just don't understand one thing) So I have a homework problem where I got a 3x6 matrix, and I have to parametrize the equations and solve for each variable in ...
5
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1answer
85 views

$A = B\cdot p(A)$. Show $A$ and $B$ commute.

A problem my professor sent out: Suppose $p$ is a polynomial with constant term nonzero. Suppose $A,B\in M_n(\mathbb{C})$ such that $A=B\cdot p(A)$. Show that $A$ and $B$ commute. This is a ...
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Solving system of linear equations

Consider $5x+3y=4$ and $3x+6y=1.$ List the set of primes for which this system of linear equations does not have a solution in the field $Z_p.$
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Linear Algebra - four “true or false” questions about matrices and linear systems

I'm reviewing for my linear algebra course, and have four "true or false" questions that I'm struggling to prove. I've included my approach to the solutions in brackets below them: 1) If $A^2 = B^2$, ...
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33 views

Summation of $\sum_{k=0}^{n-1} z^k = 0$

Suppose that $z$ doesn't equal $1$ and $z^n=1$ for some integer $n>1$. Show that: $$\sum_{k=0}^{n-1} z^k = 0$$ I'm completely stuck on this. Any and all help would be appreciated.
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System of n homogeneous equations

Suppose that $z_0, z_1,\ldots, z_{n-1}$ are the $n$ distinct solutions of $z^n=1$ Consider the system of $n$ homogeneous equations in the $n$ unknowns $x_0,x_1,\ldots,x_{n-1}$ given by: $$ ...
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1answer
20 views

Is a quotient vector space of dimension 1 the kernel of a functional?

$F$ is a field and $H$ is a subspace of the vector space $F^n$ of codimension 1. Then is the quotient space $F^n/H$ the kernel of a linear functional?
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Parallel vectors in $\mathbb{R}^n$.

Def: We say that $\vec{x},\vec{y}\in\mathbb{R}^n$ are parallel vectors if $|\vec{x}\cdot \vec{y}|=||\vec{x}||\,| |\vec{y}||$. (i.e equality holds in Cauchy–Schwarz inequality) I'm having some ...
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1answer
30 views

Proof the change of variables theorem by volume comparison

My books prove the change of variables theorem by admitting a lemma (it says that linear algebra is needed so the proof won't be listed in the book): Let $\Psi:O\to \mathbb R$ be a smooth change of ...
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31 views

Existence of a smooth curve with specific properties

Let $\gamma: [0,T] \to \mathbb R^n$ be a differentiable curve with the property that for any $t_0 < t_1$ and any vector $v \in \mathbb R^n$ $$\langle \gamma(t), v \rangle \ne 0, \;\;\text{ for all ...
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How to Find Frame with lower dimension for $C^n$

Let ${f_k}$ be a frame for $C^n$ with unit norm and frame lower bound A>1. Let I(index set) be subset of {1,2,...,m} such that $|I|<A$, where $m$ is the dimension of the frame. then ${f_k}$ where ...
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Is this matrix associated with an arbitrary group of events positive semi-definite?

Now I have an arbitrary group of events $X_1,X_2,\ldots,X_m$(with no independence or correlation assumptions, nor distribution knowledge), and define a symmetric matrix $\mathbf{K}$ as below: $$ ...
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2answers
24 views

How to show CBC = I and CAC is a diagonal matrix for B positive definite and A positive semi definite?

How would you accomplish this: Show that if $A$ is a positive semi definite matrix and $B$ is a positive definite matrix, both $n\times n$, then there is a matrix $C$, also $n\times n$, such that ...
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1answer
30 views

Find a basis of $M_2(F)$ so that every member of the basis is idempotent

Let $V=M_{2\times 2}(F)$ (the space of 2x2 matrices with coefficients in a field $F$). Find a basis $\{A_1,A_2,A_3,A_4\}$ of $V$ so that $A_j^2=A_j$ for all $j$. My attempt. Let $A_j$ be ...
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1answer
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How to show that these two versions of Farkas lemma are equal?

One version of Farkas lemma is that Let $A$ be a real $m\times n$ matrix and $b$ an $m$-dimensional real vector. Then, exactly one of the following statements are true. There exists an ...
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Is an elementry abelian group a non-degenerate symplectic vector space?

Let $A$ be an elementry abelian group with $|A|=p^{n}$ where $p$ is a prime number and $n$ is even. It is well-known that we can consider $A$ as a vector space of dimension $n$ over the field $F_p ...
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Is this an Alternative Proof of a set of vectors forming a basis?

This is one of my exam past paper question So I proved this correctly by following the normal method which is showing that a, b and c are linearly independant My proof - When I looked at the ...
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1answer
176 views

Defining an inner product from a norm which satisfies parallelogram law

Suppose we define inner product on complex inner-product space as the following : $$ \langle u,v\rangle =\frac{\|u+v\|^2 - \|u-v\|^2 + \|u+iv\|^2i - \|u-iv\|^2i}{4}$$ Given that the norm satisfies ...
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2answers
102 views

Prove that $||u+v||^2 -||u-v||^2 = 4(u\cdot v)$

Prove that $\|u+v\|^2 -\|u-v\|^2 = 4(u\cdot v)$ where u and v are vectors in R^n Edit: The subtraction part basically the second half, previously the math was incorrect. My answer: Write u = ...
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Solve the linear system by Gauss - Jordan elimination

$$ \begin{align} x& - y + 2z - w &= -1\\ 2x& + y - 2z - 2w &= -2\\ -x& + 2y - 4z + w &= 1\\ 3x& -3w &= -3 \end{align} $$ ...
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Solving this matrix

I have no idea why I am having so many issues solving this matrix, but I am. I keep getting a negative answer for y, but they are all positive. The answer should be x= 65, y= 30, and z= 45. I had ...
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1answer
375 views

Given an implicit 3D plane, how do I find the orthogonal projection matrix - which projects any point - onto this plane?

The plane is given by the equation $Ax+By+Cz+d = 0$. Can you tell me how can I figure out the 4x4 matrix which orthogonally projects any point given by homogeneous coordinates onto this plane? I am ...
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1answer
32 views

Linear Algebra Problem Proof

I have been stuck on this problem for quite some time now and, unfortunately, appear to have given up. Perhaps the minds on this page will help me out. Given an $n\times n$ matrix D, where ...
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Find all solutions of $Ax = 0$ in parametric vector form

How can I find all solutions of $Ax = 0$ in parametric vector form where A is row equivalent to the matrix $\begin{pmatrix} -1&-4&0&-4\\2&-8&0&8 \end{pmatrix}$
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Is u in the subset of $R^3$ spanned by the columns of A?

u = $\begin{pmatrix} 4\\-1\\4 \end{pmatrix}$ and A = $\begin{pmatrix} 2&5&-1\\0&1&-1\\1&2&0 \end{pmatrix}$ How can I determine if u is in the subset of $R^3$ spanned by ...
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1answer
218 views

Find a vector equation and parametric equations or the line in R^3 that passes through the point (1,2,-3) and is parallel to the vector u=(4,-5,1).

Find a vector equation and parametric equations or the line in $\mathbb{R}^3$ that passes through the point $(1,2,-3)$ and is parallel to the vector $u=(4,-5,1)$. Find two points on the line that are ...
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1answer
23 views

Determine if A is a linear combination of B when a free variable exists

$A = \begin{pmatrix} 1&0&5\\-2&1&-6&\\0&2&8 \end{pmatrix}$, $B = \begin{pmatrix} 2\\-1\\6 \end{pmatrix}$ From $A$ and $B$ I have created an augmented matrix ...