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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Equality involving exterior product..

suppose you have a differential form $\omega$ writting in local coordinates as $$\omega=\sum_{i=1}^ndx_i\wedge dy_i.$$ Can anyone help me showing the following equality: $$\omega^n=n!(dx_1\wedge ...
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63 views

Does this form of matrix have a name?

I'm looking for the name of this kind of $n$-by-$n$ matrix: $$\left(\begin{array}{cccc} -s_1 & b_{12} & b_{13} & b_{14} \\ b_{21} & -s_2 & b_{23} & b_{24} \\ b_{31} & ...
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3answers
96 views

Showing that two real matrices are not congruent over $\mathbb{Q}$

Maybe it is a stupid question but I will still ask it here. How can I prove that the following matrices are not congruent over $\mathbb{Q}$? \begin{pmatrix} -1 & 0\\ 0 & 2\\ ...
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0answers
90 views

Given a symmetric bilinear map $B:\mathbb{R}^8 \times \mathbb{R}^8 \to \mathbb{R}$ and a subspace $U,\ \dim{U}=7$ then $rk(B|_U)\geq6$

Given a symmetric bilinear map $B:\mathbb{R}^8 \times \mathbb{R}^8 \to \mathbb{R}$ and a subspace $U,\ \dim{U}=7$ with the signature $(5,3)$ then $rk(B|_U)\geq6$? Meaning if we look at $B|_{U \times ...
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1answer
220 views

Orthogonal complement of orthogonal complement

Let U be a subspace of V (where V is a vector space over C or R). The orthogonal complement of the orthogonal complement of U is not equal to U in general (equal only for dim V finite). Can anyone ...
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1answer
60 views

A matrix decomposition

Suppose $N$ is a symmetric matrix, then show that it can be uniquely decomposed as $N=N^+-N^-$, where $N^+$ and $N^-$ are both nonnegative-definite, i.e. their eigenvalue are non-negative, and ...
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3answers
53 views

Help on a proof of dimension of a vector space

The proof shows that two bases have the same number of elements, and I can't understand one step. The proof goes: As $v_1, . . . , v_n$ is a basis of $V$, each $w_k$ can be expressed as a linear ...
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3answers
95 views

Is $S=\{(a+b,a+c,2c)\mid a,b,c\in \mathbb{R}\}$ a subspace of $\mathbb{R}^3$?

Is $S=\{(a+b,a+c,2c)\mid a,b,c\in \mathbb{R}\}$ a subspace of $\mathbb{R}^3$? I just made this question up to practice determining if sets are vector subspaces or not. From what I can tell, the ...
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0answers
21 views

Data estimation based on progression

Given a data-set $x$ and $y$. x | y ------------------ 153,000 | 0.058848 332,641 | 0.36352 506,629 | 0.53 If $x$ being the number of database records ...
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0answers
114 views

Multi-affine function

Suppose I have a three-variable function $f(x_1, x_2, x_3)$, $f : \mathbb{R}^3 \to \mathbb{R}$. If it is linear for $x_1$, $x_2$ and $x_3$ we can say it has the form $f(x_1, x_2, x_3) = c_1x_1 + ...
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86 views

Proving differentiable mapping is onto

Let $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ be a differentiable mapping such that there is a $C > 0$ satisfying $C|x-y| \leq |f(x) - f(y)|$ for all $x,y \in \mathbb{R}^n$. Prove that $\det ...
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3answers
81 views

Let $x, y, z$ be vectors in a vector space $V$ such that $x+y+z=0$. Show that $\def\sp{\operatorname{span}}\sp(x, y) = \sp(y, z)$.

How do I incorporate $\def\sp{\operatorname{span}}\sp(x,y)=\sp(-y-z,y)=\sp(y,z)$ into a proof?
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1answer
78 views

Why do elementary steps of elimination appear to fail for a system of equations with each term of degree 2?

I apologise in advance if this is irritating due to its simplicity - I have only done High School Mathematics. Given: (1) $(a+1)^2$ = $(b+2)^2$+$(c+3)^2$ (2) $(a+3)^2$ = $(b+1)^2$+$(c+2)^2$ (3) ...
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0answers
102 views

Image of the Sylvester matrix is the degree of the GCD

Let $P_k(F)$ denote the $F$-vector space of (univariate) polynomials of degree $\leq n$. Letting $F$ be a field lets everything be monic, but it seems sufficient to consider a ring $R$ such that the ...
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2answers
256 views

exponential matrix

Hi i am trying to understand the exponential matrix: When is exponential matrix function $e^{At}$ integrable where A is an $n \times n$ matrix and $t$ is an $n$-dimensional vector? By integrable i ...
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2answers
36 views

If $\{u_1,\ldots,u_n\}$ is a complete orthonormal set for the inner product induced by $G$, then $\sum_{j=1}^n u_ju_j^T=G^{-1}$

For $G$, an $n\times n$ and symmetric positive definite matrix, the $G$-inner product on $\mathbb{R}^{n}$ is given by $$(x,y)_G=x^{T}Gy.$$ A complete $G$-orthonormal set ...
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0answers
161 views

How to find the parity check matrix for 101101101101101 in Hamming Codes (15,11) in graphic way?

I am trying to find hamming matrix for safe coded word: 101101101101101 My questions are: 1) What matrix check I should use? I mean there are two types of 15,11 => one starting with 1111 and one ...
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2answers
65 views

Is $S=\{(1,t)\mid t\in \mathbb{R}\}$ a subspace of $\mathbb{R}^2$?

My professor introduced subspaces of $\mathbb{R}^n$ today and I don't think I understand them very well. He posed this question as an example: Is the set $S=\{(1,t)\mid t\in \mathbb{R}\}$ a ...
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4answers
761 views

Geometric multiplicity of repeated Eigenvalues

I am still finding it difficult to determine the geometric multiplicity for repeated eigenvalues and the resultant eigenspace. For example, I am not quite sure what to do with the following matrix, ...
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1answer
127 views

Linear multiple-variable function

I'm reading a differential equation book but i'm stuck on his definition of linear ODE. Supposing our equation has y as its dependent variable, t as its independent variable and y^(n) reffers to the ...
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1answer
83 views

The dimension of the derivative

Suppose $m > 1$. Let $f: \mathbb{R}^n \rightarrow \mathbb{R}^m$ be a smooth map. Consider $f + Ax$ for $A \in \mathrm{Mat}_{m\times n}$. Define $F: \mathbb{R}^n \times \mathrm{Mat}_{m\times n} ...
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0answers
44 views

Matrices product by using packed storage

I have the following question, and I hope it won't be off-topic. Within a MATLAB code I generally have an $N\times N$ symmetric distance matrix Distance=\begin{array}{cccccccc} d_{1,1} & d_{1,2} ...
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1answer
79 views

Linear equation System

What are the solutions of the following system: $ 14x_1 + 35x_2 - 7x_3 - 63x_4 = 0 $ $ -10x_1 - 25x_2 + 5x_3 + 45x_4 = 0 $ $ 26x_1 + 65x_2 - 13x_3 - 117x_4 = 0 $ 4 unknowns (n), 3 equations ...
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1answer
169 views

Finding Matrix of improper rotation

I'm struggling with this excercise: Give a Matrix $A\in O(3)$ which is describing an improper rotation with an angle of $\pi /3$ and axis $(1,1,1)$. What do I need to do?
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1answer
119 views

The derivative of a linear transformation

Suppose $m > 1$. Let $f: \mathbb{R}^n \rightarrow \mathbb{R}^m$ be a smooth map. Consider $f + Ax$ for $A \in \mathrm{Mat}_{m\times n}, x \in \mathbb{R}^n$. Define $F: \mathbb{R}^n \times ...
2
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1answer
40 views

Sum of products of elements in matrix form.

Suppose I have two matrices $\textbf{A}$, and $\textbf{B}$ as follows: $\begin{array}{c=c} \textbf{A} = \left[ \begin{array}{ccc} a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23}\\ ...
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4answers
321 views

How to determine the rank and determinant of $A$?

let $A$ be $$A_{a} = \begin{pmatrix} a & 1 & 1 & 1 \\ 1 & a & 1 & 1\\ 1 & 1 & a & 1\\ 1 & 1 & 1 & a \end{pmatrix}$$ How can I calculate the rank of $A$ ...
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1answer
594 views

Least Squares Plane using Matricies

For a Least Squares solution to a 2D set of coordinates, the formula is: $X^T\,X\,\vec b = X^Ty$ (where $X^T$ denotes $X$ transpose) (for: $y = B_0 + B_1x + B_2x^2$) where: ...
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1answer
334 views

Determinant of Matrix Computed by Expanding Down the Diagonal?

Going through this paper: http://dx.doi.org/10.1016/S0893-9659(00)00169-5 at the bottom of page 407, the authors seem to compute the determinant of a matrix by expanding down the diagonal. The ...
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1answer
114 views

Definition of submatrix [closed]

Let $A$ a matrix, I need the definition of sub matrix of $A$. Thanks in advance.
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1k views

How do I know if the linear system has a line of intersection?

I was wondering how can I determine if there is a line of intersection with any matrix? For example, if I have the following matrix: $$\left(\begin{array}{rrr|r} 1 & -3 & -2 & -9 \\ 2 ...
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2answers
221 views

Existence of eigenvalues for self-adjoint maps in finite-dimensional inner product spaces

For a finite-dimensional inner product space over $\mathbb{C}$, it is clear that every linear transformation is diagonalisable. In my lecture notes, the lecturer claims that: For a ...
6
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1answer
189 views

Dual space of a dual space = What?

Please help me understand this bit of linear algebra! Suppose $V$ is a real vector space. Then $V^*$ --- its dual space --- is the vector space of linear maps $V\to \mathbb R$ How then do I ...
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2answers
33 views

Let T : $\Bbb{R}^n\to \Bbb{R}^n$ be the linear operator with $T(e_i) = (1,…,1)$ for all $i = 1,\ldots,n$. Find an eigenbasis for $T$.

I know that if the matrix is an $n\times n$ matrix, the eigenvalues will be $n$ with alg. multiplicity $1$ and $0$ with alg. multiplicity $n-1$. I am having a hard time generalizing the eigenbasis ...
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1answer
25 views

Find $c$ which makes $cA$ is an orthogonal projection on $A$

$A=\begin{pmatrix} 2&-1&-1\\-1&2&-1\\-1&-1&2\end{pmatrix}$ $c>0$ and $B=cA$. Find $c$ which makes $B$ is an orthogonal projection on $A$. Hmmm.....I first find the ...
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1answer
285 views

Finding the smallest subset of a set of vectors which contains another vector in the span

Consider a set $S=\{ \underline{v_1},\dots , \underline{v_n} \} $ of vectors of dimension $d<n$. Suppose for some vector $\underline{b}$ that the solution space for the matrix equation $\left[ ...
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1answer
61 views

Standard Matrices for Linear Transformation

I'm not able to find an explanation for finding the standard matrix for a linear transformation of equations. For example, if I have; $$w_1=2x_1-3x_2+x_4$$ $$w_2=3x_1+5x_2-x_4$$ Would the standard ...
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5answers
291 views

If $\dim V=v$ and $\dim(\ker T)=n$, prove that $T$ has at most $v-n+1$ distinct eigenvalues

Let $T:V\to V$ be a linear operator. If $\dim V=v$ and $\dim(\ker T)=n$, prove that $T$ has at most $v-n+1$ distinct eigenvalues. I have been working on this proof for a few days and I am not ...
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2answers
1k views

Domain, Co-Domain, and Linearity of Linear Systems homework check.

I am asked to find the domain, co-domain, and to determine whether of not the transformation is linear. I'm not sure if I am doing this properly, so I figured I would ask as my textbook doesn't have ...
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2answers
1k views

Is there NO solution to this linear system of 3 equations, $3$ unknowns?

I have the following linear system: $$\begin{align} &x + y + 2z + 2 = 0 \\ &3x - y + 14z -6 = 0 \\ &x + 2y +5 = 0 \end{align}$$ I immediately noticed that there was no $z$ term in the ...
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1answer
46 views

Linearly independent subset under linear 1-to-1 transformation

Suppose that $T:\mathbb{R}^n \to \mathbb{R}^m$ is linear and one-to-one. Let $\{v_1, v_2, \ldots, v_k\}$ be a linearly independent subset of $\mathbb{R}^n$. Prove that the set $\left\{T(v_1), T(v_2), ...
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38 views

Suppose that $T: \mathbb R^2 \to \mathbb R^2$ is the linear transformation that rotates a vector by 90°.

Suppose that $T: \mathbb R^2 \to \mathbb R^2$ is the linear transformation that rotates a vector by 90°. (a) What is the null space of T? (b) Is T one-to-one? (c) What is the range of T? ...
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1answer
80 views

How to prove the properties derived from a matrix's signature

We've recently learned about metric signatures following the proof of Sylvester's law of inertia but we didn't quite say which properties does the signature of a given matrix $A\in ...
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2answers
52 views

Confusion related to smoothness of a function

I just found this thing that $\operatorname{trace}(AB)$ where $A$ and $B$ are two matrices, it is a smooth function. I didn't understand how it is a smooth function. Any suggestions?
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1answer
48 views

Let $Q$ is the matrix with the column vectors from orthogonal basis $\beta$

Let $Q$ be a matrix such that the column vectors form an orthogonal basis $\beta$={$v_1,\dots,v_n$} of $V$. Let $\alpha$ be the standard ordered basis of $V$. Then $[I]_\alpha^\beta=Q$. I think it is ...
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1answer
298 views

Unitarily equivalent?

I'm confused about that notion. In my textbook there are two examples. (1) $A=\begin{pmatrix} 1&1&0\\0&2&3\\0&0&3\end{pmatrix}$ and $B=\begin{pmatrix} ...
0
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1answer
100 views

Elementary Matrices Row Operation

I just want to know how to get elementary matrices using fast and efficient way to solve it. Since I'm new in linear algebra, I hope someone able to help me Given $$ A =\begin{pmatrix} ...
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1answer
107 views

What's special about characteristic 2?

I'm trying to get the big picture of how bilinear forms and quadratic forms relate over fields $F$ with $char(F) = 2$ and fields with $char(F) \neq 2$. What I gather so far is that if $char(F) \neq ...
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1answer
69 views

Find a vector from a given vector transformation matrix [closed]

if anyone can give an explanation on how to solve 10.f, that would be great, thanks! Find an: $$\vec x$$ such that: $$T(\vec x) = \begin{bmatrix}2\\1\\3\end{bmatrix}$$ The whole question is here ...
2
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2answers
361 views

The second largest eigenvalue for Perron-Frobenius matrix

The Perron-Frobenius theorem is about the largest eigenvalue and eigenvector of a non-negative (irreducible) matrix. My question: Is there any estimation of the difference between the first and ...