Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them, including systems of linear equations, bases, dimensions, subspaces, matrices, determinants, traces, eigenvalues and eigenvectors, diagonalization, Jordan forms, etc. For questions specifically ...

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Help understanding the range and kernel of a linear transformation

I'm having some trouble understanding the Range and Kernel of a linear transformation. The definition goes as follows: Let $T:V \longrightarrow W$ be a linear transformation. Define the sets ...
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congruent matrices

If the rank of a matrix A is 1, the matrix is row equivalent to one with only one row different to 0. But not necessary the matrix A is congruent to one wuth only one row different to 0/ How to find a ...
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2answers
33 views

Straight line equation is linear or not?

I read somewhere that for the linearity the equation should pass through the origin in this regard the equation of straight line y=mx+c is linear or not?
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1answer
13 views

derivative of gradient involving inverse of matrices

I need to take three partial derivatives of this squared mahanalobis distance with respect to these three matrices: $Q, A,$ and $S$ $$(x+Ab)^T(A^TQA+S)^{-1}(x + Ab)$$ $x$ and $b$ are vectors of ...
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1answer
30 views

Do positive-definite matrices always have real eigen values?

Do positive-definite matrices always have real eigenvalues? I tried looking for examples of matrices without real eigenvalues (they would have even dimensions). But the examples I tend to see all ...
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7 views

Exponential Demand Periodic Review

I have exponentially distributed demand data and I am trying to find a formula for an 'order up to level (OUL)' periodic review ordering policy. We are not using a re order point for this policy. ...
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15 views

A inquality in matrix norm

Let $A,I \in {M_n}$($I$ is identity matrix) and $\left| {\left\| . \right\|} \right|$ is matrix norm.Suppose $\left| {\left\| A \right\|} \right| < 1$ and $\left| {\left\| I \right\|} \right| \ge ...
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1answer
17 views

Prove every isometry on an odd- dimensional real product space has 1 or -1 as an eigenvalue.

This is a question from Axler. I was hoping for some help. It seems easy to understand, but I don't know where to go about on proving this.
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16 views

Questions about the position of a matrix and a vector

I got quite confused by position of a matrix and a vector. For example, the definition of a range space put a matrix in front of a vector, like $R(A) = \{ Ax | x∈ R^n \}$; However for a linear ...
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23 views

Positive linear functionals on the space of positive semidefinite matrices

Suppose $f: \mathbb{R}^{n \times n} \rightarrow \mathbb{R}$ is a linear functional with the property that $f(A) \geq 0$ whenever $A$ is positive semidefinite. Is it true that there exist vectors $v_1, ...
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1answer
37 views

Is there a function over $\mathbb{Z}_p$ that is never linear?

Let $p$ be a prime. I wonder if there is a function $f$ that satisfies the following rule: Whenever $$z_1 + \dots + z_c \equiv cx \mod p$$ (where $1 < c < p$, and $z_j, x \in \mathbb{Z}_p$) ...
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1answer
16 views

Clustering of vectors via inner product relationship

This might be an odd question, but suppose I have a lot of vectors $a_i\in\mathbb{R}^{3}$ (not necessarily unit) and for some unit vector $u\in\mathbb{R}^{3}$ I find $$ \sum_{i=1}^m ...
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1answer
19 views

How to find the span for a linear transformation?

I'm learning Linear Transformations and I understand what is a linear transformation. Now I'm trying to look at an example question and I'm not really sure how the span is found. The question goes as ...
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4answers
39 views

Prove that $\|v \|^2= |\langle v, e_1 \rangle |^2 + \cdots + | \langle v, e_m\rangle |^2$

Suppose $(e_1,\cdots, e_m)$ is an orthonormal basis in $V$. Let $v \in V$ . Prove that $\|v\|^2= |\langle v, e_1 \rangle |^2 + \cdots + | \langle v, e_m\rangle |^2$ Let $v\in V$ and ...
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21 views

What is wrong with my solution to this problem?

The base $ABCD$ of the figure has area $9$. The point $M$ divides the segment $AB$ on ratio $2$ and the edge $BF$ of length $2$ forms an angle of $60º$. Calculate $[CM,CB,BF]$, knowing that ...
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5 views

How many binary vectors of weight 3 can you have before their span contains one of weight 2?

In other words, I am looking for the smallest $k$ for which the following is always true: Let $v_i \in \mathbb{F}_2^n$ for $i = 1\ldots k$ be distinct vectors of Hamming weight 3, that is, vectors ...
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0answers
13 views

Behavior of MGF of Quadratic Combination of Dependent Multivariate Gaussians

Sorry if the formatting is poor, this is my first time asking a question. I'm investigating how squared gaussians behave, using the techniques provided here, which are giving me inconsistent results. ...
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1answer
39 views

An Extension to the Generalized Eigenvalue Problem

Given two square matrices $A_1,A_2 \in \mathbb{R}^{n\times n}$, the generalized eigenvalue problem is finding the scalar $\lambda \in \mathbb{C}$ and vector $x \in \mathbb{C}^{n}$ such that $$ ...
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Algebra Word Problem Help 1337 [duplicate]

Jamal borrowed $\$7,000$ to buy a used car. He borrowed some of the money from a bank that charged $7.8\%$ simple interest and the rest from a friend who charged $9\%$ simple interest. If the total ...
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2answers
30 views

Let $V$ be a $K$ - Vector Space, $f: V \rightarrow V$ a linear map. Under some condition, show that $v, f(v),…,f^n(v)$ is linear independent.

Let $V$ be a $K$ - Vector Space, $f: V \rightarrow V$ a linear map. Let $v \in V$. May a number $n ≥ 0$ exist, so that: $f^n(v) \not= 0$ and $ f^{n+1}(v) = 0$. Show that $v, f(v),...,f^n(v)$ is ...
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Bayesian statistics and Basis for continous functions

I was thinking about Bayesian statistics, and one thought bothered me: In Bayesian statistics, we assume that the pdf $p(x)$ can be described as: $p(x)=\int f(x|\theta)g(\theta)d\theta$ usually ...
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1answer
21 views

Norm of a complex cross product

Let $c=(c_1,c_2,c_3)$ be a complex vector. How can we see that $\|c\|^2=\|c\times \bar{c}\|$? Here the bar means component wise complex conjugation, the norm is the Hermitian norm, and the cross ...
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Algebra Word Problem 1337 [on hold]

Jamal borrowed $\$7,000$ to buy a used car. He borrowed some of the money from a bank that charged $7.8\%$ simple interest and the rest from a friend who charged $9\%$ simple interest. If the total ...
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3answers
39 views

Uniqueness of basis vectors

Say I have 2 vectors $v_1$ and $v_2$ as basis of a subspace. Then is it true that $kv_1$ and $mv_2$ where $k$ and $m$ are real numbers, are also basis for that subspace?
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17 views

How would we generate a basis of sigmoidal functions?

I am trying to figure out how to generate a basis of sigmoidal functions. My issue is thus: there are several possible generating functions for a sigmoid (logistic curves, error functions, arctangent, ...
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1answer
30 views

Let A and B be n*n matrices such that trace(A)<0<trace(B).

Let A and B $n\times n$ such that trace(A)$\lt0\lt$trace(B). Then, $f(t)=1-det(e^{tA+(1-t)B})$ has 1) infinitely many zeros in $0\lt t\lt1$ 2) at least one zero in $\Bbb R$ 3) no zeros 4) either ...
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3answers
44 views

Are $\cos$ and $\sin$ linearly dependent in $[- π , π]$? [on hold]

Are $\cos$ and $\sin$ linearly dependent on $[- π , π]$ If true, demonstrate; if False show a counterexample.
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1answer
28 views

I have to show center of $M_n(H)$ is $\mathbb{R}I$ [on hold]

Let $H$ be a real quaternion ring. I have to show that the center of $M_n(H)$ is $\mathbb{R}I$. Can anyone help?
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3answers
198 views

How do I restrict k to ensure my matrix has exactly 3 distinct eigenvalues?

$$A=\begin{bmatrix}-1&-1&0\\-12&3&-1\\k&0&0\end{bmatrix}$$ How do I restrict $k$ to ensure that my matrix has 3 distinct real eigenvalues? I tried going about it the long way ...
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Endomorphism commutes with its adjugate

Let $R$ be a commutative ring, $M$ a free $R$-module of rank $n$ and $f \in \rm{End}(M)$. The adjugate $f^\sharp$ of $f$ is defined by the equalities $$ f^\sharp(x) \wedge y = x \wedge ...
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2answers
46 views

Distance between points

Suppose I have two matrices each containing coordinates of $m$ and $n$ points in 2 D. Is there an easy way using linear algebra to calculate the euclidean distance between all points (i.e., the ...
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0answers
43 views

A question in matrix norm [on hold]

Let $I,A \in {M_n}$ and suppose $\left| {\left\| . \right\|} \right|$ be a matrix norm $\left| {\left\| I \right\|} \right| \ge 1$ and $\left| {\left\| A \right\|} \right| < 1$($I$ is identity ...
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1answer
49 views

Determinant proof using its properties

Prove without expanding: \begin{equation} \begin{vmatrix}bc&a^2&a^2\\b^2&ac&b^2\\c^2&c^2 & ab\end{vmatrix} = ...
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2answers
55 views

Matrix ring $M_2(\mathbb{C})$, $\mathbb{C}^2$ with $M_2(\mathbb{C})$-module structure.

Let $R$ be the $2 \times 2$ matrix ring $M_2(\mathbb{C})$. let $M = \mathbb{C}^2$ with its natural $R$-module structure (just given by the usual action of $2 \times 2$ matrices on $2$-dimensional ...
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0answers
20 views

How to make use of symmetric of sparse matrix to solve this kind of problem?

I have the following matrix to be solved: $$\left\{ \matrix{ {a_{11}}{x_1} + {a_{12}}{x_2} + \cdots + {a_{1n}}{x_n} = {y_1} \hfill \cr {a_{21}}{x_1} + {a_{22}}{x_2} + \cdots + {a_{2n}}{x_n} = ...
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1answer
17 views

Dependence of product of matrix and a vector, on the rank of a Matrix

What is the significance of the rank of a matrix, say $A$, when I am multiplying a vector, say $x$, by $A$? In other words, let $x$ be a column vector of suitable dimension and let $rank (A)=m$. What ...
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3answers
81 views

$\frac{1}{{1 + {\left\| A \right\|} }} \le {\left\| {{{(I - A)}^{ - 1}}} \right\|}$

Let a matrix norm $ {\left\| . \right\|}$ have the property that $ {\left\| I \right\|} = 1$ and $ {\left\| A \right\|} < 1$. Why does the following inequality hold? $$\frac{1}{{1 + \left\| A ...
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0answers
43 views

How to find one matrix, which is subject to $B^3 = A$. How much is such matrices? [duplicate]

Here I have a problem with row echelon form. $$A := \begin{bmatrix}-6 & 3 & 7 \\ 0 & -1 & 0 \\ -14 & 12 & 15\end{bmatrix}$$
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1answer
26 views

Invertibility for a matrix that I don't know [on hold]

I would like to know why $(e^{-At}-I)^{-1}$ is invertible when matrix A is Hurwitz.
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1answer
26 views

Show that the matrix is a symmetric matrix

Let $T:V\to V$ be a symmetric linear map i.e $\langle Tx,y\rangle =\langle x,Ty\rangle $ .$V$ is a finite dimensional inner product space If $\{e_i:1\leq i\leq n\}$ is an orthonormal basis of $V$ ...
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3answers
27 views

Determinant of symplectic matrix

A $2n \times 2n$ matrix $S$ is symplectic, if $SJ_{2n}S^T=J_{2n}$ where \begin{equation} J_{2n} = \begin{bmatrix} 0 & I_n \\ -I_n & 0 \end{bmatrix}. \end{equation} My question is, how to ...
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6answers
75 views

How to prove there's a vector $z \in \mathbb{R}^4$ orthogonal to two linearly independent vectors $x,y \in \mathbb{R}^4$?

Let $x, y \in \mathbb{R}^4$ with $\{x, y\}$ being linearly independent. Prove that there exists a non-zero vector $z$ that is orthogonal to both $x$ and $y$. Any hints on what to do after the ...
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2answers
35 views

If Q is an orthogonal matrix, does it follow that $QDQ^T = Q^TDQ$?

Say A is a real, $n \times n$ symmetric matrix. Then it is orthogonally diagonalisable, with $A = QDQ^T = QDQ^{-1}$. Let's say we do not know that Q is symmetric (at first) - does the above hold?
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3answers
91 views

Can systems of 3 linear equations with 3 unknowns have more than one solution?

In each part,determine whether the given vector is a solution of the linear system \begin{align} 2x-4y-z&=1\\ x-3y+z&=1\\ 3x-5y-3z&=1 \end{align} (a) $(3,1,1)$ (b) $(3,-1,1)$ (c) ...
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25 views

Is $\oplus_{i \in I} M_i$ necessarily projective?

If $\{M_i\}$ is a collection of projective modules over a ring $R$, then must the direct sum $\oplus_{i \in I} M_i$ necessarily be projective? I understand that each direct sum of two modules is ...
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1answer
13 views

Invertibility of Product implies invertibility of factors

Say $C=AB$ where $A,B,C$ are all $n\times n$ matrices. It's easy to show that if $A$ and $B$ are invertible then $C$ is invertible --> $C^{-1}=B^{-1}A^{-1}$. Does the converse hold? That is, if $C$ ...
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1answer
41 views

$\left\| {\left| {BA - I} \right|} \right\| < 1$ $ \Rightarrow $ $A$ and $B$ are both nonsingular

Let $A,B \in {M_n}$ satisfy the inequality $\left\| {\left| {BA - I} \right|} \right\| < 1$ and $\left\| {\left| . \right|} \right\|$ be a matrix norm on ${M_n}$.Why do $A$ and $B$ are both ...
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1answer
19 views

reordering the indices of a matrix

Let $A$ be an $n \times n$ matrix of rank r. Then by reordering the indices if necessary we can bring the matrix in the form $(\frac{A_1}{A_2})$ where $A_1$ is an $r \times n$ matrix, $A_2$ is an $n-r ...
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3answers
47 views

Let {v1, v2} be a basis for a subspace S of R 3 . If B = {w1, w2, w3} is a set of vectors in S, then B cannot be linearly independent.

Let $\{v_1, v_2\}$ be a basis for a subspace $S$ of $\Bbb R^3$ . If $\mathcal B = \{w_1, w_2, w_3\}$ is a set of vectors in $S$, then $\mathcal B$ cannot be linearly independent. I'm not sure how ...
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3answers
31 views

Use Vectors To Show Three Vertices Belong to a Right Triangle

The Full Question Theorems Used This is what I call theorem 1: My Work This problem has two major steps as far as I can see. First, I must show that these are points of a triangle(not ...