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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Extending a subset of $\mathbb{F}_q^n$ to a basis

So I have a linearly independent subset of $\mathbb{F}_q^n$ and I'm trying to extend it to a basis. I heard this can be done, but I can't find an algorithm for actually doing it. Can you help me? ...
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126 views

Confusion with bilinear form and linear operation relation.

My lecture notes say that for every bilinear form there exists a linear operator such that $$\tau (v,w) = v.(Tw)$$ and that there must exist some other linear operator $S$ such that $$(Sv).w = ...
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86 views

The exact definition of an orthonomal basis?

In the area of bilinear forms, my lecture notes say that there is a basis $\{e_i\}$ of $V$ with respect to which $\tau (e_i, e_j) = \delta_{ij}$ where $$\delta_{ij} = \begin{cases} 1 & i=j \\ 0 ...
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555 views

What matrices preserve the $L_1$ norm for positive, unit norm vectors?

It's easy to show that orthogonal/unitary matrices preserve the $L_2$ norm of a vector, but if I want a transformation that preserves the $L_1$ norm, what can I deduce about the matrices that do this? ...
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148 views

Exercise at the Beginning of Part II in Fulton's Book on Young Tableaux

In Fulton's Book Young Tableaux, there's an Exercise at the beginning of part II for which I cannot find a solution (there doesn't seem to be one for this exercise in my copy of the book). It reads: ...
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3answers
364 views

Matrix multiplication proof: $(A+B)^k$ for matrices such that $A^2=B^2=0$ and $AB=BA$

I have the following details about the matrices $A,B$: $$A_{n\times n},B_{n\times n}$$ $$A^2 = B^2 = 0$$ $$AB=BA$$ I need to find $x \in \mathbb{N}$ so $(A+B)^x=0$ What implications can I make ...
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608 views

Every norm on $\mathbb{R}^n$ is equivalent to $\|\cdot \|_\infty$?

We have just come across the lemma that every norm on $\mathbb{R}^n$ is equivalent to $\|\cdot \|_\infty$. My question is that do all norms on $\mathbb{R}^n$ take the form $\|\cdot \|_1, \|\cdot \|_2, ...
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312 views

Definition of non-singular quadratic form

A quadratic form is a homogenous polynomial $q(x_1, \dots , x_n) = \sum_{j=1}^n \sum_{i=1}^n x_i x_j a_{ij}$. Let $A := (a)_{ij}$. We define $q$ to be non-singular (or non-degenerate) if its ...
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280 views

What is the relation of basis in linear algebra and basis in topology?

In linear algebra and topology ,it all has the concept basis,but I can not construct the relation of them,could you explain the relation of two basis,such as the basis in linear algebra is special ...
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376 views

On gluing regular tetrahedra together to form a ring

The above are part of the articles and some background of it. And there is one claim in the articles, saying that " in order to show that no trivial ring can be formed, it is sufficient to show ...
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671 views

Is the dual space of a subspace of a vector space simply its orthogonal complement?

Is the dual space of a subspace of a vector space simply its orthogonal complement? My professor seems to use the terms interchangeably, but from Wikipedia they seem to be quite distinct concepts ...
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1k views

Vector direction given the points?

I have the following image: I want to know the direction of orange arrow given the two points, I know that if the direction is (0, -1) the orange will be pointing up, but how can I get the ...
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144 views

Is it always true that $\|\vec{x} + \vec{y}\| \geq \|\vec{x}\| - \|\vec{y}\|$ over $\mathbb{R}^n$?

If $\vec{x}, \vec{y} \in \mathbb{R}^n$. Is it always true that $ \|\vec{x} + \vec{y}\| \geq \|\vec{x}\| - \|\vec{y}\| $ ? Any advice or proofs would be greatly appreciated.
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147 views

“Projection” in normed vector space

Just out of curiosity, is the following true? If $U$ is a subspace of a normed vector space $X$, and $x\in X\setminus U$, then $$\inf_{u\in U}\|x-u\|>0.$$ If it is true, is it moreover true ...
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76 views

The homomorphisms $\mathbb{Z}^n \to G$ for finite abelian $G$

I can't get my head around the fact that for any finite abelian group $G$ of size $k$ there are $k^n$ homomorphisms $\mathbb{Z}^n \to G$. It's meant to be 'clear' but can someone please explain why?
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281 views

Solving polynomial matrix equations over finite fields

The concrete problem is this: Find triplets of distinct matrices $(A,B,C)$ of dimension $6\times 6$ over the field $\mathbb{F}_{2^2}$ such that: $A^2B=AB^2$ $C^2A=CA^2$ $B^3C=BC^3$ However, I'm ...
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517 views

Generic topology on a vector space?

For a (possibly infinite-dimensional) vector space $V$, I thought about the following topology $\tau$: Let $O \in \tau$ if every $x \in O$ has the property that for every $v \in V$, there is an ...
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141 views

Resolving the up-vector of an image through multiple transforms

First up, a disclaimer. I'm an engineer not a mathematician, so my approach thus-far may be a little ham-fisted. I have however reached a point in this problem where my usual engineering references ...
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517 views

Linear Algebra Working with Linear Transformations

Let $v_1=[-3;-1]$ and $v_2= [-2;-1]$ Let $T:\mathbb{R}^2 \rightarrow \mathbb{R}^2$ be the linear transformation satisfying: $T(v_1)=[15;-6]$ and $T(v_2)=[11;-3]$ Find the image of an arbitrary ...
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337 views

Elementary row operations effect on determinants

I have a matrix $$ A = \begin{pmatrix} 1 & 2 & 3 & 4\\ -1 & 1 & 2 & 3\\ 1 & -1 & 1 & 2\\ -1 & 1 & -1 & 1\\ \end{pmatrix} $$ I should be ...
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35 views

Dimensions of various subspaces

I am given that $A$ is a $12\times 15$ matrix and the equation $Ax = b$ has a solution for every $b \epsilon \mathbb{R}^{12}$. What are the dimensions of the domain, the range and the kernel of $A$ I ...
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556 views

Basic logic problem with verbal question, confirmation whether right or wrong

Problem 2a here on page 882, translated Prove the statement If $\lambda\in \sigma(A)$, so $\lambda^p \in\sigma(A^p) \forall p\in\mathbb N.$ (where $\lambda$ is an eigen-value and ...
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92 views

Decision problem for existence of arbitrage portfolio

Given $m$-dimensional vector $p$ (price vector), and $m\times n$ matrix $X$ ($m$ securities' payoffs in $n$ states) for arbitrary $m$ and $n$, is there an algorithm to decide if there exists $h\in ...
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80 views

If $A\ge0$ and $B\le 0$, are the eigenvalues of $AB$ non-positive?

Let $A\in\mathbb{R}^{n\times n}$ be a symmetric positive semi-definite matrix, and $B\in\mathbb{R}^{n\times n}$ is symmetric negative semi-definite. How to prove the eigenvalue of $AB$ is either zero ...
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82 views

Confusion with eigenvalues, needs clarification

Problem 3 on page 882 here, translated: The matrix $A=\begin{bmatrix} 1 & 1 & 0 \\ 0 & 2 & 0 \\ 2 & 1 & -1 \\ \end{bmatrix}$ has eigen-vectors $\begin{bmatrix} 1 ...
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42 views

What is the space of relations?

I was reading the current issue of AMM when I came across this term "space of relations", which I don't understand. Basically, we are given 9 vectors (0,1,0,0), (2,0,0,0), (1,1,0,0), (3,0,0,0), ...
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68 views

The decomposition of inner product space

If $V$,$W$are two inner product spaces and $L:V\to W$ is a linear map with its adjoint $L^\star$, then is there a decomposition of $W$=ker$(L^\star)$ $ \oplus $ im$(L)$ ? (It is easy that the ...
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Use of determinants

I have been teaching myself maths (primarily calculus) throughout this and last year, and was stumped with the use of determinants. In the math textbooks I have, they simply show how to compute a ...
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338 views

Eigenvalues of a sum of a non-negative symmetric Toeplitz and a non-negative diagonal matrix

I have an $n\times n$ Toeplitz matrix $\mathbf{A}$ that is non-negative and symmetric (that is, $A_{i,j}=A_{j,i}=a_i\geq 0$) and a diagonal matrix $\mathbf{B}=\operatorname{diag}(b_1,b_2,\ldots,b_n)$ ...
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361 views

Get the direction vector passing through the intersection point of two straight lines

Let say I have this diagram, How to find the direction vector passing through the intersection point of two straight lines? Update: new vector is the bisector of two lines and vector may be ...
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1answer
633 views

Probability of full rank of a random matrix.

Suppose, $G$ is a $k \times n$ binary matrix with $\operatorname{rank}(G) = k$. The first $k$ columns of $G$ are linearly independent and the next $n-k$ columns are linear combinations of the first ...
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175 views

number of invertible 0-1 matrices

Define the set $S_n=\{A_n| A_n \hbox{is invertible 0-1 matrix}\}$. What is the size of $S_n$? When $n=2$, it is easy to see $\sharp S_2=6$. I guess $\sharp S_n=\prod_{k=1}^n(2^n-2^{k-1})$.
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160 views

Show/Prove two conditions when $P$ is an $n \times n$ matrix such that $P^2 = P$ and $P^t = P$.

Let $P$ is an $n \times n$ matrix such that $P^2 = P$ and $P^t = P$. With this I am supposed to do two show/prove two conditions: Let $E = \operatorname{Col}(P)$. Show that $E^{\perp} = ...
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71 views

Show that $v \cdot w = 0$ under specific conditions.

Let $S$ be a symmetrix $n \times n$ matrix. (Remember that means $S^t = S$). Let $v$, $w$ be eigenvectors of $S$ for eigenvalues $\lambda$, $\mu$ respectively. Suppose $\lambda \not= \mu$. Show that ...
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121 views

If $\alpha_{1},\ldots,\alpha_{m}$ are vectors different from zero vector, then there is a linear functional $f$ such that $f(\alpha_{i})\neq 0$

I am self-studying Hoffman and Kunze's book Linear Algebra. This is exercise 14 from page 106. Let $\mathbb{F}$ be a field of characteristic zero and let $V$ be a finite-dimensional vector space ...
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2k views

Find a subset that forms a basis for the span of a set

Let $p_1 = 1 - 2t - t^2$, $p_2 = t + t^2 + t^3$, $p_3 = 1 - t + t ^3$ and $p_4 = 3 + 4t + t^2 + 4t^3$. Let $S$ be the set of these four functions. Find a subset of $S$ that is a basis for the span ...
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3k views

Free variables in echelon form.

If free variables are the variables that aren't basic variables, then how did this example come up with x4? Reduced echelon form: ...
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331 views

The real part of a matrix under similarity transformation

I have a question regarding the real part of some matrix A, defined as $$ Re\{A\} = \frac{1}{2}\left(A + A^\dagger \right).$$ Where $A^\dagger$ denotes the Hermitian conjugate. One can also assume ...
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257 views

Spectral radius, and a curious equality.

Given a $N\times N$ matrix $A$ over $\mathbb R$. Let $ \rho\left( A \right) = \max \left\{ {\left| \lambda \right|;\lambda \mbox{ eigenvalue of }A} \right\}$. Someone told me that, the following ...
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160 views

conjugate function for matrix mixed norm

I am familiar with the conjugate function of the vector norm, which uses the concept of dual norm and is defined as follows: ...
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290 views

Bijection between bilinear forms and linear operators

My lecture notes say that: ''Given any bilinear form $\tau$ on $V$, there is a uniquely determined linear operator $T$ on $V$ such that $$\tau(v,w) = v \, .(Tw)$$ so once we've fixed a 'starting' ...
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96 views

Positive definite symmetric bilinear forms on $\mathbb{R}$

A Euclidean space is defined to be a vector space $V$ over $\mathbb{R}$ together with a positive definite symmetric bilinear form $\tau$. My notes say that if we assume that $V$ is a Euclidean space ...
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522 views

How to solve Linear Diophantine equations?

I have read about Linear Diophantine equations such as $ax+by=c$ are called diophantine equations and give an integer solution only if $\gcd(a,b)$ divides $c$. These equations are of great importance ...
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189 views

Positive definite bilinear form question

I'm given that a quadratic form $q$ is positive definite if $q(v) > 0 \quad \forall 0 \neq v \in V$ and equivalently for bilinear form $\tau$. Does this mean that in a positive definite bilinear ...
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134 views

Kernel of the Lie bracket

Let $\mathfrak{g}$ be a dimension 3 Lie algebra and $[\quad,\quad]$ be a rank 1 map from $\bigwedge^{2}\mathfrak{g} \rightarrow \mathfrak{g}$. In this case, the kernel of $[\quad,\quad]$ is $3 - 1 = ...
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186 views

Eigenvalues/vectors as measures of 'frequency'

This is a question about an off-hand remark from a lecturer a few weeks ago. He was talking about eigenvectors/values of a matrix, and rhetorically asked us if we'd seen the interpretation of ...
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174 views

Calculate the new length of line in Rectangle after chanaging Rectangle width and height

Let say I have the following image, How to calculate the new length of line? Update: It seems there is confusion on question. See this image, I just need to map the 100 * 100 Rectangle with 250 ...
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1answer
611 views

rotating a rectangle via a rotation matrix

I want to rotate a 2D rectangle using a rotation matrix. After the rotation, I want the points (x, y) of the rectangle to be: ...
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105 views

Problem on the determinant of a Matrix

Suppose we have $n-1$ linearly independent vectors $a_1, \ldots,a_{n-1} \in \mathbb{Z^n}$. Is it possible to find another vector $a_n\in \mathbb{Z^n}$ such that the determinant of the matrix $M$ ...
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167 views

Is $f:M_n(\mathbb{C})\longrightarrow M_n(\mathbb{C})$ continuous?

I want to know whether this is absurd question or reasonable to ask: Let $f:M_n(\mathbb{C})\to M_n(\mathbb{C})$ be given by $f(A)= B$, where $B$ is a diagonal matrix having the same eigenvalues as ...