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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2
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3answers
86 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) ...
0
votes
0answers
8 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 ...
1
vote
1answer
26 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 ...
1
vote
0answers
20 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, ...
0
votes
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.
0
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0answers
6 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. ...
1
vote
1answer
750 views

Find 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 a vector equation and parametric equation of 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 ...
0
votes
0answers
13 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 ...
0
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0answers
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 ...
0
votes
1answer
34 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 $z_j, x \in \mathbb{Z}_p$) ...
1
vote
1answer
18 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 ...
1
vote
1answer
7k views

Magnitude of a Matrix?

Consider a vector V. The magnitude of this vector (if it describes a position in euclidean space) = distance from the origin is simply: $(V^TV)^{1/2} $ aka the square root of the dot product... ...
6
votes
0answers
54 views

Let $\mathbb{K} $ be a field of characteristic $p>0$ and $\mathbb{F} | \mathbb{K} $ a finite and separable extension.

Let $\mathbb{K}$ be a field of characteristic $p>0$ and $\mathbb{F}/ \mathbb{K}$ a finite and separable extension. Show that if $B=\{\alpha_1,\dots,\alpha_n\}$ is a basis, then ...
1
vote
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 ...
0
votes
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 ...
0
votes
1answer
33 views

column space of a matrix

If $A\in M_{m\times n}\mathbb{(R)}$, show that $\mathcal{R}(AA^t)=\mathcal{R}(A)$ and $\mathcal{R}(A^tA)=\mathcal{R}(A^t)$ where $\mathcal{R}$ denotes the column space of matrix. How can I prove it ...
0
votes
1answer
33 views

Strategies for linear systems

Consider I have the following equations. Is there a faster way for me to solve the system without going through a series of substitutions? $$-20a+13b+13c=0$$ $$10a-26b+13c=0$$ $$10a-13b-16c=0$$ ...
0
votes
0answers
20 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 ...
30
votes
5answers
19k views

How do I tell if matrices are similar?

I have two $2\times 2$ matrices, $A$ and $B$, with the same determinant. I want to know if they are similar or not. I solved this by using a matrix called $S$: $$\left(\begin{array}{cc} a& b\\ ...
0
votes
0answers
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 ...
3
votes
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. ...
3
votes
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 $$ ...
1
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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 ...
1
vote
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 ...
0
votes
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?
-2
votes
0answers
23 views

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 ...
0
votes
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 ...
0
votes
0answers
19 views

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 ...
4
votes
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 ...
1
vote
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?
-5
votes
0answers
39 views

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 ...
2
votes
1answer
529 views

Determining the ratios needed in gear reduction

I am trying to work out the math behind building a gear box for turning a gear a specific RPM from a small motor. Given that a typical DC hobby motor turning at 200 RPM, and a target in the final ...
1
vote
0answers
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, ...
0
votes
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 ...
-4
votes
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.
4
votes
1answer
66 views

problem book on linear algebra

please refer a problem book on 1.Linear algebra :TOPICS: Vector spaces, Linear dependence of vectors, basis, dimension, linear transformations,matrix representation with respect to an ordered basis, ...
3
votes
3answers
197 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 ...
1
vote
0answers
52 views

Can a matrix be similar to more than one matrix?

I have a little query about similar matrices I've been struggling with. Suppose I have a 5x5 diagonal matrix A with 5 distinct eigenvalues as entries in the main diagonal. The question is, to how ...
1
vote
1answer
19 views

Calculating the determinant of an iterationmatrix

Let $C_\omega = (I-\omega D^{-1}L)^{-1}((1-\omega)I+\omega D^{-1}R)$ then $\det(C_\omega) = (1-\omega)^n$ (Where $C_\omega\in \mathbb{R}^{n\times n}$, $R$ is upper triangular, $L$ is lower ...
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votes
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 ...
2
votes
0answers
24 views

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 ...
1
vote
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 ...
1
vote
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 ...
0
votes
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} = ...
3
votes
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} = ...
0
votes
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.
0
votes
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 ...
0
votes
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. $$A := \begin{bmatrix}-6 & 3 & 7 \\ 0 & -1 & 0 \\ -14 & 12 & 15\end{bmatrix}$$
2
votes
1answer
53 views

A determinant coming out from the computation of a volume form

I am convinced that the following identity is true: \begin{equation} \det\begin{bmatrix} 1+a_1^2 & a_1 a_2 & a_1 a_3 & \ldots & a_1a_n \\ a_1a_2 & 1+a_2^2 & a_2a_3 & ...
3
votes
3answers
51 views

A vectorspace over an infinite field is not a finite union of proper subspaces?

Show that if V is a vector space over an infinite field F, then V cannot be written as set-theoretic union of a finite number of proper subspaces.