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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337 views

Prove that if its reduced row echelon form is [R c] then R is the reduced row echelon form of A.

Let $[A\;b]$ be the augmented matrix of a system linear equations. Prove that if its reduced row echelon form is $[R\;c]$, then $R$ is the reduced row echelon form of $A$. How do I prove it? I mean ...
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1answer
73 views

find the coordinates from a distance matrix [duplicate]

I want to determine whether there exists $5$ points in $\mathbb R^4$ such that the following matrix is the distance matrix. $$ \begin{pmatrix} 0& \sqrt5 & \sqrt5 & \sqrt5 & \sqrt5 ...
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1answer
85 views

See if system is linear

I have this system $$y_{3} [n]= 2x[n-2]x[n-3]$$ and I have to see if the system is linear. I understand that I have to see if the system verifies the additivity and homogeneity properties. I'm having ...
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1answer
56 views

Linear Independence of an infinite set 2. [duplicate]

Let $e_n=\sin nx$ ($x\in [-\pi,\pi])$and let $A=\{e_i|i\in \mathbb{N}\}$. Prove that A is a linearly independent set. Some hours back I had posted this question. In the meantime I was trying myself ...
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1answer
129 views

Elementary lower-triangular $4\times 4$ matrices

What are the three elementary lower triangular $4 \times 4$ matrices and what does their operation do? How can I prove that for all of these, $\det(L)=1$ and $L(x)^{-1}=L(-x)$?
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3answers
75 views

How to show $T$ is not one-one and $T$ is not ont0?

Suppose $V$ is the space of all $n \times n$ matrices with real elements. Define $T : V \to V$ by $$T (A) = AB − BA,\; A \in V,$$ where $B \in V$ is a fixed matrix. Show that for any $B \in V$, ...
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2answers
67 views

If $A$ and $B$ are $n\times n$ matrices, prove that $|(A^TB)|^2\leq|A^TA||B^TB|$; when is this an equality?

Let $A$ and $B$ be square $n$-matrices. Prove that $|(A^TB)|^2\leq|A^TA||B^TB|$. Also, under what circumstances are the left and the right side equal? I've tried multiple times, both sides should ...
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2answers
174 views

Elements in $\text{GL}(n,q)$ with irreducible characteristic polynomial

Let $x,y\in\text{GL}(n,q)$ be of the same order such that both the characteristic polynomials of $x,y$ are irreducible. Must $x,y$ always be conjugate in $\text{GL}(n,q)$? More restrictly, I am ...
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1answer
2k views

How to find a transformation matrix having several original points and their respective transformed results?

I have three original points $pt_1, pt_2, pt_3$ which if transformed by an unknown matrix $M$ turn into points $gd_1, gd_2, gd_3$ respectively. How can I find the matrix $M$ (all points are in ...
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1answer
42 views

Definition of normal subgroup vs normal operator

Why is a normal operator defined as T such that $T^*T=TT^*$ rather than as an element in some normal subgroup N, i.e. $N = \{n| n\in G, gng^{-1}\in N\}$. Is the idea that a normal operator is "normal" ...
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2answers
173 views

Linear Independence of an infinite set .

Let $e_n=\sin nx$ ($x\in [-\pi,\pi])$and let $A=\{e_i|i\in \mathbb{N}\}$. Prove that A is a linearly independent set.
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2answers
290 views

What does it mean for a subspace to be stable?

I'm looking through a proof for a spectral theorem, but I can't figure out what it means for a subspace to be stable. $\dots \mathbb{C}v$ is $T$-stable (for some $v$ that is an eigenvector of an ...
2
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3answers
46 views

Matrix decomposition again

If some matrix (M×N) can be expressed as product of (M×1) and (1×N) vectors: what is proper term for such kind of decomposition? how to tell if such kind of decomposition exists for given matrix? ...
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1answer
282 views

Tensor and Wedge Product of Vectors

I have a little doubt about tensor product acting on vectors. I was reading Spivak's Calculus on Manifolds, and Spivak introduces the tensor product of multilinear functionals. Latter he introduces ...
3
votes
1answer
652 views

tensor product with dual space

I will explain what I know, and then I will ask my question. Let $V$ and $W$ be vector spaces such that at least one is finite dimensional. In class, we showed that if either $V$ or $W$ is finite ...
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1answer
61 views

What is a “rotated” basis?

My text (p. 19) introduces the concept of a "rotated" basis without explanation. What properties or characteristics of a basis make it "rotated" with respect to another? What operation on one basis ...
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2answers
74 views

A basis for a vector space

Prove that for any non-zero linear function $f$ of a $n$-dimensional vector space $V$ there exists a basis ${e_1, ..., e_n}$ for the $V$ vector space, if $$f(x_1e_1+ ... + x_ne_n) = x_1$$ should be ...
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1answer
81 views

Lower bound on the norm of product of non square matrices

The following inequality is known: $\parallel AB\parallel\geq\parallel A\parallel \sigma_{n}(B)$. However, it is only valid where both $A$ and $B$ are square. Is there an analogue for rectangular ...
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1answer
272 views

Powers of Irreducible Transition and Periodic Transition Matrices

Suppose P is irreducible transition matrix with period d. How many communicating classes does P^k have and what is the period of each state?
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1answer
69 views

Overdetermined system of equations

$x_{1}=u+v$ $x_{2} = \zeta u + \zeta^{2} v$ $x_{3} = \zeta^{2} u + \zeta v$ where $\zeta = -\frac{1}{2} + i \frac{\sqrt{3}}{2}$ Since $\zeta$ is a third root of unity, $\zeta^{3} =1 $. Also, ...
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1answer
60 views

Find point $X$ such that line through plane $E$ and sphere $S$ meet at $(0,0,1)$ (stereographic projection)

Find the point $X$ such that the line going through the plane $E$ and sphere $S$ meet at the point $(0,0,1)$ (stereographic projection). Let $S$ denote the unit sphere $$S = \{(x,y,z) \in ...
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2answers
94 views

Proof by induction on $k$

Can anyone give me some pointers on where to start on this question. I have tried to do a base step for $k=1$ but its just too complicated and I don't know what to try: $$\int_0^x t^ke^{\lambda ...
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3answers
228 views

Can a non-zero vector have zero image under every linear functional?

Let $X$ be an infinite-dimensional vector space, and let $x_0$ be an element of $X$ such that $f(x_0)=0$ for every linear functional $f$ defined on $X$. Then can we prove that $x_0$ is the zero vector ...
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1answer
53 views

Linear congruency

please I need some help here... (dont understand the answer) it's a task on congruency. I have the feeling that at the end something went worng. is it possible to have a negative solvation? Thank ...
4
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1answer
122 views

Find out whether a matrix is a distance matrix or not

I have a $5 \times 5$ symmetric matrix $A$ with zeroes on the diagonal and I am supposed to find whether there exist 5 points in $\mathbb R^4$ such that $A$ is the distance matrix. How can I solve ...
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2answers
63 views

What can a linear transformation do in $\mathbb{R}^2$?

If I have points of a unit circle (centered at an origin) $$ \left\{ \left. \begin{pmatrix} \cos(\varphi) \\ \sin(\varphi) \end{pmatrix} \right| \varphi \in [0;2\pi) \right\} $$ and I affect them ...
3
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2answers
279 views

How to prove that two non-zero linear functionals defined on the same vector space and having the same null-space are proportional?

Let $f$ and $g$ be two non-zero linear functionals defined on a vector space $X$ such that the null-space of $f$ is equal to that of $g$. How to prove that $f$ and $g$ are proportional (i.e. one is a ...
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2answers
158 views

a multiple choice question on non-negative definite matrices

A symmetric matrix in $\mathbb{M}_n(\mathbb{R})$ is said to be non-negative definite if $x^TAx≥0$ for all (column) vectors $x \in \mathbb{R}^n$. Which of the following statements are true? a. If a ...
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10answers
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Coordinate-free proof of $\operatorname{Tr}(AB)=\operatorname{Tr}(BA)$?

I am searching for a short coordinate-free proof of $\operatorname{Tr}(AB)=\operatorname{Tr}(BA)$ for linear operators $A$, $B$ between finite dimensional vector spaces of the same dimension. The ...
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2answers
54 views

assume $A\in M_n(\mathbb C)$,$A^2=0$ how prove $\exists C,B\in M_n(\mathbb C)$ such that A=BC and CB=0?

assume $A\in M_n(\mathbb C)$,$A^2=0$ how prove $\exists C,B\in M_n(\mathbb C)$ such that $A=BC$ and $CB=0$. Thanks in advance
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0answers
90 views

Sorting combinations of linearly independent vectors

Given a set of $m$ vectors in $\mathbb{R}^n$ ($m > n$), sort all combinations of $n$ linearly independent vectors according to the determinant of the matrix whose columns are the $n$ vectors. ...
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1answer
308 views

How to diagonalize this matrix?

Consider the $n\times m$ matrix $M=[M_1, \ldots, M_m]$ where the $i$-th column reads $$ M_i= \,^t(\underbrace{1,\ldots,1}_{a_i},0,\ldots,0) $$ where the $a_i$'s are given positive natural numbers. ...
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1answer
73 views

L : $\mathbb{R}^n \rightarrow \mathbb{R}^m$ is a linear mapping, linear independence of $L$ mapped onto a set of vectors.

Here is my question: Let $L:\mathbb{R}^n\to \mathbb{R}^m$ be a linear mapping. Assume that $\{\vec{v}_1,\ldots,\vec{v}_n\}$ is a basis for $\mathbb{R}^n$ such that $\{\vec{v}_1,\ldots,\vec{v}_k\}$ is ...
2
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1answer
985 views

Inverse of orthogonal projection

I have an $N \times N$ orthogonal projection matrix $P = A^H(AA^H)^{-1}A$ that I'm trying to find the inverse for. I'm using matlab, however, I keep getting the warning "the matrix is close to ...
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4answers
1k views

Do $ AB $ and $ BA $ have same minimal and characteristic polynomials?

Let $ A, B $ be two square matrices of order $n$. Do $ AB $ and $ BA $ have same minimal and characteristic polynomials? I have a proof only if $ A$ or $ B $ is invertible. Is it true for all ...
3
votes
1answer
180 views

Linear Algebra- reflection matrix

$A$ is reflection matrix $2\times2$. $$B=A^4-2A^3-A-5I.$$ Find numbers $k$, $t$ in $\mathbb R$ so that $B^{-1}=kB+tI$. I know that reflection matrix have eigenvalues of $1$, $-1$ ($A^2=I$) I got ...
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2answers
273 views

$(n-1)$-dimensional subspace of an $n$-dimensional vector space

Let $f$ be a fixed nonzero linear functional on an $n$-dimensional vector space $V$ and $H=\{\alpha \in V:f(\alpha)=0\}$. Then $H$ is a subspace of $V$ and its dimension is $n-1$. I have shown ...
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3answers
98 views

Extending a functional on a subspace to one on the whole vector space

The dual space of a vector space $V$ is the vector space of all linear functionals on $V$. Denote the dual space of $V$ by $V'$. Question: If $W$ is a subspace of a finite dimensional ...
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3answers
64 views

equivalence relation (even number)

I have two problems with this task. First I can´t do the correct mathematic shape. And also when is an equivalence it´s supposed to be reflexive, symmetrical and transitive. The first two are okay. ...
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3answers
77 views

A simultaneous system of equations

Solve for $a,b,c$: \begin{align} 2ab+a+2b=24\\ 2bc+b+c=52\\ 2ac+2c+a=74\\ \end{align} Solving them simultaneously is leading to very difficult situation. Plz help.
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2answers
341 views

Solve a quadratic matrix equation?

Given a known symmetric matrix $M$, vector $\vec{v}$ and scalars $a$ and $b$, I'm trying to solve for a scalar $x$ such that: $\vec{v}^T(M+(ax+b)I)^{-1}\vec{v} - ...
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2answers
155 views

Internal direct sum of vector spaces

Let $U$ be the subspace of $\mathbb{R}^3$ spanned by $\{(1,1,0), (0,1,1)\}$. Find a subspace $W$ of $\Bbb R^3$ such that $\mathbb{R}^3 = U \oplus W$. As I am having an examination tomorrow, it would ...
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1answer
47 views

Proof of a linear transformation property

Suppose $\phi:X \rightarrow Y$ is a map of sets and $F$ is a field. Let $\phi^* : F(Y) \rightarrow F(X)$ be a map sending a function $f \in F(Y)$ to a function $\phi^*(f) \in F(X)$ given by ...
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2answers
78 views

Implicit function theorem : statement and the rank of the matrix

This is perhaps something standard in linear algebra(a subject I am somewhat weak in). So I apologize in advance. I would be grateful if someone can guide me. The statement of implicit function ...
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1answer
46 views

$A\vec u =2\vec u$ and $A\vec{v}= 3\vec{v}$, are $\vec{u}$, $\vec{v}$ linearly dependent?

The Question: Let $A$ be an $n$ x $n$ matrix and $\vec{u}\mbox{,} \vec{v} \in \mathbb{R}^n$ such that $A\vec{u} = 2\vec{u}$ and $A\vec{v} = 3\vec{v}$. $\vec{u} \ne \vec{0}$ and $\vec{v} \ne \vec{0}$. ...
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3answers
22 views

Summation of non-null value [closed]

Sorry, am a bit rush of time, the main question in the picture.. How should I express the situation in the picture in a concise expression? Summation doesn't help as it does not skip the null value. ...
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1answer
164 views

Is unitary $T^2$ if $T$ is unitary?

If $T$ is unitary then $\|Tx\|=\|x\|$ then $T^2=T T$, then $\|Tx\| \|Tx\| =\|x\| \|x\|$ $\|Tx|\|^2 =\|x\|^2|$ but $T^2$ is unitary because $T^2=T T$ and $T$ was unitary then $\|Tx\|=\|x\|$ Is true ...
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0answers
80 views

Inner product and orthogonal base

Let $V=C^2$ and $\alpha= (x_1,x_2)$ and $\beta= (y_1,y_2)$ two elements of $V$. Let $g$ the form on $V$ defined by ...
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1answer
82 views

Dependence relations on the rows of a matrix

Can anyone help with finding the dependence relations on the rows? I know how to do it for the columns but a bit stumped with rows. I know it has something to do with a book-keeping matrix tho.
7
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1answer
210 views

Pathological linear functionals and ZF

Let $S$ be an infinite set. Let $C(S)$ be the vector space of all functions $S \to \mathbb{R}$, and let $C_c(S)$ be the subspace of functions of finite support. Is the existence of a nonzero linear ...