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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Prove an inequality in a group ring

Let $$G=\bigoplus_{n\in\mathbb{Z}}\left(\mathbb{Z}/2\mathbb{Z}\right)_n$$ be a group, and for any $n\in \mathbb{Z}$, denote $\delta_n$ to be the element in $G$ with $n$-th coordinate $1$ and zero at ...
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1answer
1k views

An eigenvector is a non-zero vector such that…

Various sources define eigenvalues and eigenvectors in slightly different ways (context independent). For example, both of the following definitions seem not to exclude the zero-vector as an ...
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5answers
925 views

Linearly independent set can be completed to a basis

Suppose I have a linear independent set for a finite dimensional vector space $V$. How can I prove rigorously than it can be completed to a basis? More importantly, is the completion unique? I do not ...
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1answer
92 views

Singular values of a diagonal matrix concatenated with a vector?

If $\Sigma$ is a diagonal matrix and $\vec{x}$ a vector, is there a simple formula for the singular values of the matrix $[\Sigma, \vec{x}]$? A hint: first, write out our matrix $[\Sigma,\vec{x}]$ ...
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2answers
743 views

det $A = $ det $B$ and det $U= 1\,$ , so $\,A = UB$

Assume that $A$ and $B$ are invertible $n\times n$ matrices. Prove that $\det A=\det B$ if and only if $A=UB$, where $U$ is a matrix with $\det U=1$. Showing that $A = UB$ implies equal determinants ...
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3answers
521 views

When does the kernel of a function equal the image?

When does the kernel of a function equal the image? Thanks in advance
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4answers
99 views

Contradiction in the rules regarding determinants and row operations?

In my textbook it says that if you multiply a row in a matrix $A$ by a nonzero constant $c$ to obtain $B$, then $\det{B}=c\det{A}$. Later on it says that if you obtain $B = cA$ by adding $c$ times ...
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1answer
113 views

Can we find subgroup of $(\mathbb{R},+)$ with order 2?

We used the following idea: first get a set of Hamel basis for $\mathbb{R}$, secondly, divide it into two parts such that one set of the Hamel basis forms a group, the other one is just the former one ...
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3answers
567 views

norm of a linear operator

On http://www.proofwiki.org/wiki/Definition:Norm_(Linear_Transformation) , it is stated that $||Ah|| \leq ||A||||h||$ where $A$ is an operator. Is this a theorem of some sort? If so, how can it be ...
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1answer
54 views

If $Tr(A^k)=0$ for $1\leq k\leq n$ is $A^n=0_n$, where $A\in M_n(\mathbb R)$?

I know that the relation $Tr(A^k)=0$ is equivalent to $\sum_1^n(\alpha_i^k)=0$ where $\alpha_i$ is an eigenvalue of A and that $\alpha_i=0$ for any i is a solution to the system, but is it unique? ...
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26 views

Two subspaces $A$, $B$ are paralel, if and only if they are both subspaces of a space with a dimension equal $d = 1 + \max (\dim A, \dim B)$?

It is true, that two affine subspaces $A$, $B$ are paralel, if and only if there exists an affine space $C$, that both are subspaces of $C$, and $\dim C = 1 + \max (\dim A, \dim B)$? If so, do You ...
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4answers
1k views

Finding the intersection of two lines in terms of y and x

I am trying to understand how I can find the point of intersection between two lines numerically. I know how to do this graphically. I would need the two equations in slope intercept form and then ...
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2answers
187 views

Expanding $(I+A)^{-1}$

I think that there is an expansion of $(I+A)^{-1}$ when $A$ is a matrix with a norm smaller than 1, but I cannot seem to recall this expansion. Any suggestions?
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1answer
115 views

Permutation Matrix

I need to find a Permutation matrix, for an E matrix, i have to permute it because i need to use always two of the rest of eigenvalues of matrix A to operate with them, matrix A is numerically defined ...
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1answer
62 views

What is the probability that a random sparse vector lies in a subspace?

Let $\mathbb{F}$ be a Finite Field. For $m\leq n$ a vector $v$ in $\mathbb{F}^n$ is $m$-sparse if $ \sum_i (v_i \neq 0) \leq m$, i.e. , the hamming weight is almost m. Let we call $S(n,m)$ the set of ...
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0answers
61 views

'Injectivity' of a bilinear map restricted to the set it can generate starting from a given vector

Here is a problem I'm stuck with: Let $V$ be a vector space (on a field $F$) of finite dimension $n$, $v\in V$ and $\mu : V\times V \mapsto V$ a bilinear application determined by its action on the ...
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2answers
61 views

How is the function “determinant” well-defined and where is recursion theorem used?

Let's first denote $\overline{A}_{1j}$ to be a matrix where first row and $j$-th column are removed from $A$. Definition of determinant, as usual, is: $\det(A)=\sum_{j=1}^n (-1)^{1+j} A_{1j} ...
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2answers
241 views

Uniqueness of the Jordan decomposition.

I have seen it said that a matrix $M$ (over $\mathbb{C}$, say) has a unique decomposition $M=D+N$ where $D$ is diagonal and $N$ is nilpotent. I'm having trouble seeing this, since the Jordan form of a ...
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6answers
309 views

Proof that $\mathbb{R}[x]$ is not a finite dimensional vector space

How can we prove that the vector space of polynomials in one variable, $\mathbb{R}[x]$ is not finite dimensional?
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1answer
352 views

Uniqueness of sum of nilpotent and diagonalizable matrices.

I have the following question: Let $V$ be a vector space over a field $F$, and let $A$ be an endomorphism $V\rightarrow V$. Prove there is at most one pair of linear maps $D$ and $N$ such that ...
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Prove $\sqrt{3a + b^3} + \sqrt{3b + c^3} + \sqrt{3c + a^3} \ge 6$

If $a,b,c$ are non-negative numbers and $a+b+c=3$, prove that: $$\sqrt{3a + b^3} + \sqrt{3b + c^3} + \sqrt{3c + a^3} \ge 6$$ Here's what I've tried: Using Cauchy-Schawrz I proved that: $$(3a + ...
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1answer
106 views

Graphically, what is positive semidefinite-ness?

Suppose that we are trying to minimize a function $f$ on $\mathbb{R}^n$ and we apply Newton's method, updating: \begin{align} \mathbf{x}_{n+1} = \mathbf{x}_n - [\nabla^2 f(\mathbf{x}_n)]^{-1} \nabla ...
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2answers
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Algebra: sums of rows and columns given, find elements

I stumbled upon the following problem, which got me confused: \begin{array}[x]{3} ~ & y_1 & y_2\\ z_1 & x_{11} & x_{12} \\ z_2 & x_{21} & x_{22} \\ \end{array} $z_1$ and $z_2$ ...
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1answer
110 views

How to use Pythagoras theorem with alternative axes

please see the following picture and tell me if you know the solution. Thank you very much.!
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1answer
105 views

decomposing PSD block matrix into two PSD block matrices

Given $Q = \left( \begin{array}{ccc} A + B & C \\ C^T & D\end{array} \right) $, where we know that $Q, A, B, D$ are all positive semi-definite, square, but not necessarily equally sized ...
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1answer
244 views

Calculating the axis used to rotate a matrix 90 degrees

I have a problem that I can't seem to get worked out. I have two rotation matricies: $$R_1 = \left( \begin{array}{ccc} 1 & 0 & 0 & 567.057 \\ 0 & 1 & 0 & -138.337 \\ 0 & ...
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1answer
96 views

Prove that a surface of revolution is a 2dimension manifold

I have a question about surface of revolution. Prove that a surface of revolution is a 2dimension manifold.
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45 views

multiplication of positive matrix and non negative matrix

Assume $A>0$ and $H\le0$ are two real matrices I need to prove that $\mbox{trace}(AH)\le0$ I tried to diagonalize $A$ with an orthogonal matrix but I didn't get any results. Thanks.
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Finding a vector orthogonal to a subspace

Suppose you were given vectors $a_1,\dots,a_n \in \mathbb{R}^m$ then how would you compute some vector orthogonal to the given list of vectors? Note that you are allowed to return the zero vector only ...
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1answer
176 views

Matrices to model 3D object

I'm toying around with an algorithm to determine placement of 3D objects into a larger 3D space. I immediately thought of using matrices. It's been some years since my Linear Algebra courses. I was ...
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3answers
105 views

Why is a real-valued inner product defined on a Euclidean space a continuous function

I really need a real maths proof that inner product $\langle \cdot, \cdot \rangle:X \times X \to \mathbb{R}$, where $X$ is an Euclidean space, is a continuous function.
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1answer
467 views

Find the eigenvalues and eigenvectors of the line reflection in $xy$ plane

Given a $xy$ plane, if we have line like $10x+6y=0$. To find the eigenvalues and eigenvectors of its reflection. Could we say for reflection, we always have eigenvalues $0$ and $-1$? And by matrix ...
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1answer
378 views

Matrix transformation shearing the triangle to be right triangle

If we have a triangle at $(1,1), (5,3), (7,1)$, how to find the sheared matrix to transform the triangle to be right triangle at $(1,1)$. Is it that we need to find $i$ in $\begin{pmatrix} 1 ...
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3answers
811 views

If $P$ is real orthogonal matrix with $\det P = -1$, prove that $-1$ is an eigenvalue of $P$.

If $P$ is real orthogonal matrix with $\det P = -1$, prove that $-1$ is an eigenvalue of $P$. can anyone help me please how can I solve this problem?
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2answers
235 views

to show $(I_n+S)$ and $(I_n-S)$ are non-singular if $S$ is skew symmetric

If $S$ be a real skew-symmetric matrix of order $n$ , prove that the matrices $(I_n+S)$ and $(I_n-S)$ are both non-singular. can anyone help me please to solve this problem.thanks for your help.
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1answer
48 views

How to expand this matrix formula?

$L$ is a matrix, is it possible to expand this formula in terms of $\operatorname{Tr}L^n$? $$\log\operatorname{Tr}e^L$$
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0answers
35 views

diagonalise a symmetric bilinear form [duplicate]

I have a math question that I got stuck on and would like to ask you about: I have a symmetric bilinear form on $\mathbb{Q}^4$ which is described w.r.t the standard basis. $$g(v,w)=\mathbf{v}^\top ...
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1answer
33 views

let $n$ is non-negative number so $n$ is the solution for the equation

let $n$ is non-negative number so the equations -->$(x^2+1)^2+n = yz+1$ -->$(y^2+1)^2+n = zx+1$ -->$(z^2+1)^2+n = xy+1$ have $(x,y,z)$ real solution. find all solutions for the non-negative n ...
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1answer
184 views

Prove by using diagonalization

Can anyone give me some hints on how to prove this question? Q: Use diagonalization to prove that if $A \subset B$ are lattices then $[B:A ]=\frac{\Delta(A)}{\Delta(B)}$. Added: Definition: A ...
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2answers
184 views

Let $a,b,c$ are positive real numbers such that $b^2+c^2<a<1.$

I came across the following question that says: Let $a,b,c$ are positive real numbers such that $b^2+c^2<a<1.$ Consider the $3 \times 3$ matrix $A=\begin{pmatrix} 1 &b &c \\ b ...
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1answer
455 views

Determinant of the transpose of elementary matrices

Is there a 'nice' proof to show that $\det(E^T) = \det(E)$ where $E$ is an elementary matrix? Clearly it's true for the elementary matrix representing a row being multiplied by a constant, because ...
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158 views

matrix and vector multiplication distribution rules

(M x f) . (N x g) M and N are square matrices n-by-n, g are column vectors n-by-1. "." is dot product. "x" is matrix multiplication Under what conditions can I ...
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1answer
2k views

Linearly Independent set of vectors that spans the same subspace of $\mathbb{R}^3$

I'm having trouble setting this up. I have these $3$ column vectors: $\langle 1, 1, 2\rangle$ $\langle -7, -1, -8\rangle$ $\langle 3, 0, 3\rangle$ I need to find a linearly independent set of ...
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2answers
93 views

Hint for proving ${\rm SL}_n(\mathbb{R})$ is generated by matrices that are off-diagonal entries added to identity. [duplicate]

Let $S = \{ I_n + a\cdot e_{i,j} \mid a\in\mathbb{R},\ i,j= 1,\ldots,n,\ i\neq j\}$, where $e_{i,j}$ is the matrix with $1$ at entry $(i,j)$ and zero elsewhere. I need a hint to help prove that ...
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3answers
207 views

Sylow $p$-subgroups of Finite Matrix Groups

Let $G$ be a subgroup of $GL_n(\mathbb{F}_p)$, with $n\le p$, and let $P$ be a Sylow $p$-subgroup of $G$. Do all non-trivial elements of $P$ have order $p$? I believe the answer is yes, because I ...
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1answer
334 views

A linear operator always has a cyclic vector

I'm trying to prove that given a linear operator $f$ on $\Bbb{K}^2$, where $\Bbb{K}$ is a field, any non-zero non-eigenvector is a cyclic vector for $f$. Then I'm asked to show show that $f$ either ...
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2answers
227 views

Max and min value of $7x+8y$ in a given half-plane limited by straight lines?

So, there are four inequalities: $$\begin{eqnarray*} y &\geq &-3x+15; \\ y &\leq &-11/3x+56/3; \\ x &\geq &0; \\ y &\geq &0. \end{eqnarray*}$$ If we draw all those ...
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1answer
178 views

Find intersection of three planes

I am studying and my friends and I are stuck on one question that has us all thinking. It goes something like this: Give an example of three planes that only intersect at $(x, y, z) = (1,2,1)$. ...
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4answers
1k views

Two linearly independent vectors perpendicular to vector $u$

I'm having trouble with these types of questions. I have the following vector $u = (4, 7, -9)$ and it wants me to find 2 vectors that are perpendicular to this one. I know that $(4,7,-9)\cdot (x,y,z) ...
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1answer
187 views

Why is there always a Householder transformation that maps one specific vector to another?

I'm wondering why, if I'm given two vectors $u$ and $v$, I can always find a Householder transformation that maps $u$ to $v$. (This is needed in QR factorisation with Householder transformations). ...