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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Rational solutions to a system of equations

I have a system of equations $$\begin{align} xy + 3zw = 0; \\ xz + 2yw = 0; \\ xw + yz = 0. \\ \end{align}$$ Plugging it into a CAS, I see that all the rational solutions to this system have ...
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22 views

Linear polynomials relatively prime iff $ad-bc \ne 0$

Two nonzero polynomials $a+bx$ and $c+dx$ are relatively prime in $\mathbb{R}[x]$ if any only if $ad-bc \ne 0$. It's not too hard to show this on a case-by-case basis by enumerating each possible ...
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1answer
55 views

Is there anything “nice” about the set of normal matrices (over $\Bbb R$ and $\Bbb C$?)

Normal matrices are of course useful to any linear algebra buff, not least because of the spectral theorem. However, taken as a whole, they tend to have some not-so-nice properties. For example: ...
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9 views

Bounds on the singular values of a matrix with unitary columns

if $X$ is a matrix with unitary columns ( each column has unit norm ), are there lower and upper bounds on the minimum and maximum singular values of $X$? I could prove a lower bound for ...
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2answers
712 views

A subset of n vectors is linearly independent iff it spans V

I am trying to solve #8 from this PDF: http://www.math.purdue.edu/~lai37/MA353/M353P1.pdf The solutions are given in http://www.math.purdue.edu/~lai37/MA353/MP1soln.pdf I am not really seeing the ...
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1answer
19 views

Linear system associated to a translation of a subspace

Let $ X \subset R^n$ such that $ X = x_o + S$, where S is a subspace of $ R^n$ with dimension k. If $ m = n-k$, show that $\exists A \in R^{m \times n} $ and $b \in R^m$ such that $$X = \{ x \in R^n ...
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33 views

Nonhomogeneous Linear Systems and Vector Space Solutions

Are there any nonhomogeneous linear systems with a solution set that forms a vector space? I know that, in order to be a vector space, a set must consists of a set V together with operations + (called ...
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29 views

Linear Algebra: Prove by contradiction: three mutually orthogonal nonzero vectors do not exist in $\mathbb{R}^{2}$. [on hold]

Prove by contradiction: Three mutually orthogonal non-zero vectors do not exist in $\mathbb{R}^{2}$ . (Assume three vectors exist $[x_1, x_2] [y_1,y_2] [z_1,z_2]$. Show $x_1, y_1,$ or $z1$ is ...
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19 views

Performing elementary row operations on matrices

Suppose you’re doing elementary row operations on matrices which have real entries and have 3 rows. (a) Write down the elementary matrix which correspond to the elementary row operation r3 → r3 + π ...
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9 views

Efficient algorithm to find a minimum spanning set for a given vector.

A few days ago a colleague proposed the following problem. Let $W$ be a finite subset of a vector space $V$, and let $v\in\langle W\rangle$ (the linear span of $W$). Is there an efficient ...
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Algebra HELP!!!! [on hold]

Ok so this assignment is due tomorrow. Ok the question is,among the carrer home run leaders for Major League Baseball, Hank Aaron has 175 fewer than twice the number that Dave Winfield has. Hank ...
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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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23 views

Find values of h such that the vectors (2, 4) and (h, 6) span $\mathbb{R}^2$

My homework is asking me to answer problems such as the one that follows: Find all values of $h$ such that the vectors $\{a_1, a_2\}$ span $\mathbb{R}^2$, where $a_1 = (2, 4)$ and $a_2 = (h, 6)$. I ...
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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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22 views

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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1answer
15 views

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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4 views

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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1answer
21 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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1answer
66 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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350 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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2answers
18 views

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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39 views

$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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10 views

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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45 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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$\vec{a} \times \vec{b} = \vec{c} \times \vec{d}$ . 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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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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20 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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+100

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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1answer
21 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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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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17 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 ...
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$A = B\cdot p(A)$. 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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2answers
35 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
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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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34 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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33 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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19 views

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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7 views

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 ...