For questions about vector spaces and their properties. More general questions about linear algebra belong under the [tag:linear-algebra] tag. A vector space is a space which consists of elements called "vectors", which can be added and multiplied by scalars. In other words, these are the spaces ...

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18 views

How can one characterise the number of linear combinations of m > 2 linearly independent vectors that map onto the same point in the plane?

I have m > 2 vectors v in the plane, any two of which are linearly independent to each other. Any two of these vectors are enough to fill the plane. My question is this: How can one characterise ...
4
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3answers
142 views

Doubt with vectorial spaces (Basis and dimension)

Good night, i'm working in a problem, i need an basis and the dimension of the space. $a_{1}=(1,0,0,-1),\:a_{2}=(2,1,1,0),\:a_{3}=(1,1,1,1),\:a_{4}=(1,2,3,4),\:a_{5}=(0,1,2,3)$ I make this: $\left[ ...
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1answer
32 views

Uniqueness of endpoints of half-open line segments in linear spaces.

I try to solve the following exercise, which is Exercise 1.18 in Robert Megginson's An Introduction to Banach Space Theory. Let $X$ be a linear space, and define for any $x_1, x_2 \in X$ the line ...
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13answers
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Calculate Rotation Matrix to align Vector A to Vector B in 3d?

I have one triangle in 3d space that I am tracking in a simulation. Between time steps I have the the previous normal of the triangle and the current normal of the triangle along with both the current ...
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2answers
30 views

Prove that a linear mapping between vector spaces is an open mapping iff

Let $(N,|| \ ||)$ and $(N_1,||\ ||_1)$ be normed vector spaces and $f$ a linear mapping of $N$ into $N_1$. Prove that $f$ is an open mapping if and only if $\forall$ $n \in \Bbb N $, $B_r(0) \...
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0answers
30 views

Sum of projection operators

Given $p_1, ..., p_n$ $n$ projection operators on the vector space $E$ such that $\sum_{i=1}^n p_i$ is a projection operator. How to show that $\forall i,j \text{ s.t. } i \neq j, p_i \circ p_j = 0$ ?...
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0answers
25 views

mean-deviation form, why orthogonal?

This is from my textbook Why the column of the new design matrix are orthogonal? for example, let say $A=\begin{pmatrix} 1& 1& 4\\ 1& 2& 0\\ 1& 3& 2 \end{pmatrix}$ ...
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1answer
30 views

Vector norm lemma and proof

I have a question from Numerical linear algebra book by Trefethen & Bau : Let $\|\cdot\|$ denote any norm on $C^m$. The corresponding dual norm $\|\cdot\|'$ is defined by the formula $\|x\...
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0answers
24 views

Finding altitude and azimuth with an accelerometer and magnetometer

I posted this in the astronomy stack exchange forum, but considering that it is a very math intensive question I figured there could also be people on here that could help. For a project with my ...
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1answer
27 views

finding inner product

This is from my textbook: I don't know how to tell whether the spanning set are actually orthogonal. The textbook's solution is like this, forexample, to see if $P_0(t)$ and $P_1(t)$ are orthognal, ...
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2answers
420 views

Vector Project onto Subspace

So the question is: Let S be the subspace of $\mathbb{R}^3$ spanned by the vectors $ u_2 = \begin{pmatrix} \frac{2}{3}\\\frac{2}{3}\\\frac{1}{3}\end{pmatrix} u_3 = \begin{pmatrix} \frac{1}{\sqrt{2}}\\\...
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2answers
589 views

How to prove $\dim(U)=\dim(W)=\dim(V)-1 \implies V=U+W$ based on the following assumption?

Suppose $U$ and $W$ are subspaces of a vector space $V$ such that $\dim(U) =\dim(W)$ and $U\ne W$, how to prove $\dim(U)=\dim(W)=\dim(V)-1 \implies V=U+W$? My approach is to use $\dim(U+W)=\dim(U)+\...
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2answers
25 views

Find a point 90° left or right from a point (x,y,z) in a 3D space.

How can I find a point which is 90° left or right from a point (x,y,z) in a 3D space? for example if I have the point $(x,y,z)$ how to find $(x1,y1,z1)$ and $(x2,y2,z2)$.
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1answer
24 views

vector space homomorphism for $Map(\mathbb{F}_{5} , \mathbb{F}_{5})$

I'm currently stuck at a mathematical problem and I really don't know where to start.. Since I'm not an expert in Algebra over finite fields... It goes "Define a $\mathbb{F}_{5}$-vector space ...
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1answer
27 views

Intersection of normed speces and desity

Let $(X_n, \|\cdot\|_n)$ be a sequence of normed spaces. My first question is, whether it is possible to norm $X= \cap_n X_n$. My idea would be to take $\|\cdot\|_X = \sup \|\cdot\|_n$ if it is ...
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1answer
43 views

Basis for dual in infinite dimensional vector space. [closed]

I know to make it for $V$ finite dimensional. Give me a hint for infinite dimensional case. Let $V$ be a vector space with basis $\{v_{i} \}_{i \in I}$. For each $i \in I$, let $f_i \in V^{\ast}$ ...
0
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1answer
26 views

Finding span of intersection of two vector subspaces

I was trying to follow this answer, but as the comment to that answer suggests, there's a problem with dimensions, and that's exactly where I'm stuck. More concretely, I have subspaces $U$ and $W$, $...
1
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2answers
22 views

Does anti-Hermtian matrices from a vector space?

My book states that $n\times n$ anti-Hermitian matrices $T^\dagger = -T$ form a real vector space. But the identity matrix is not anti-Hermitian and hence doesn't belong to this set. Is my book wrong?
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1answer
63 views

How to compute the projection of a vector on a plane

Can someone check whether my work is correct or not? Compute the projection of $(1,1,1)$ onto the plane that passes through the points $(1,0,-1), (3,7,-3), (-2,-1,2)$. My attempt: Let $u = (1,1,...
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1answer
49 views

Dimension of subspace of $\text{End}(\mathbb{R}^5)$

I'm doing a problem which presented me with a basis for some $U\subseteq\mathbb{R}^5$ where $\dim U=3$ (I can give it explicitly if that helps but I do not think it matters). The question is this: ...
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9answers
2k views

A linear operator commuting with all such operators is a scalar multiple of the identity.

The question is from Axler's "Linear Algebra Done Right", which I'm using for self-study. We are given a linear operator $T$ over a finite dimensional vector space $V$. We have to show that $T$ is a ...
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0answers
16 views

Solution space of semilinear equation

I found the following lemma and the corollary in a paper and I don't know how to prove them. Therefore I was wondering if one of you could help me. Let $E$ be a field, $ \sigma: E \rightarrow E$ ...
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3answers
6k views

Span of an empty set is the zero vector

I am reading Nering's book on Linear Algebra and in the section on vector spaces he makes the comment, "We also agree that the empty set spans the set consisting of the zero vector alone". Is Nering ...
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0answers
11 views

Change of Coordinates and Basis

Let $P_{2}(\mathbb{R})$ denote the vector space of real polynomial functions of degree less than or equal to two and let $\beta := \{p_{0}, p_{1}, p_{2}\}$ denote the natural basis of $P_{2}(\mathbb{R}...
2
votes
1answer
18 views

If $M$ is a simple $R$-module, and an $F$-space, why does $End_F(M)\cong M^{\oplus\dim_F(M)}$?

Suppose a ring $R$ is an $F$-algebra for $F$ a field, and $M$ is a simple $R$-module and a finite dimensional $F$-vector space. We can endow $\operatorname{End}_F(M)$ with an $R$-module structure by ...
0
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0answers
42 views

Algebra quotient space homomorphism

I have to prove the following; Let $A$ be an algebra over a field $K$. If $I \subset A$ is an ideal, then there exists a unique algebra structure on the quotient vector space $A/I$ such that the ...
1
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2answers
22 views

Understanding a certain step in a proof about a basis of a vector space

This is a theorem from Roman's textbook "Advanced Linear Algebra"(p.$48$). Theorem $1.9.$ Let $V$ be a nonzero vector space. Let $I$ be a linearly independent subset of $V$ and let $S$ be a ...
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0answers
29 views

Linear algebra textbook for quantum computing?

I'm looking for an recommendation for a linear algebra textbook specifically to give me the background for learning about quantum computing, and quantum mechanics more generally. In particular, none ...
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2answers
25 views

If every non-zero vectors be the eigenvector of a real matrix $A$, prove that $A$ is the scalar matrix $\lambda I_n$.

If every non-zero vectors in $\mathbb{R}^n$ be the eigenvector of a real $n \times n$ matrix $A$ corresponding to a real eigenvalue $\lambda$, prove that $A$ is the scalar matrix $\lambda I_n$. I ...
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0answers
43 views

The set of all $n\times n$ matrices A such that the $A^T = A^{-1}$ is a subspace of the vertor space $M_n(\mathbb{R})$

I think the set of $n \times n$ matrices such that $A^T = A^{-1}$ is not a vector space since it doesn't have $0$. How do I show that it's not a subspace?
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1answer
49 views

Is the converse of the Pythagorean Theorem false for complex inner products?

I was thinking about the converse of the Pythagorean theorem: $\lVert x + y\rVert^2 = \lVert x\rVert^2 + \lVert y\rVert^2 \implies x \perp y$ Does this hold if the inner product $\langle \cdot,\...
3
votes
1answer
208 views

Mean value theorem and scalar field proof

Assume that $f′(x;y)=0$ for every $x$ in some $n$-ball $B(a)$ and for every vector $y$. Use the mean value theorem to prove that $f$ is constant on $B(a)$. And if $f′(x;y)=0$ for a fixed vector $y$ ...
0
votes
2answers
59 views

derivative of a vector

find $\frac{d^2\vec{S}}{dt}$ where $\vec{S}=(t+1)\hat{i}+(t^2+t+1)\hat{j}+(t^3+t^2+t)\hat{k}$ So $\frac{d\vec{S}}{dt}=\hat{i}+(2t+1)\hat{j}+(3t^2+2t+1)\hat{k}$ now when I take the derivative again ...
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0answers
27 views

Simple excercise on linear transformations - confused

A Linear tranformation L in $\mathbb R^3$ with matrix $$ L_b^b = \left(\begin{matrix} 1 & 0 & 5 \\ 0 & -2 & 2 \\ 1 & -2 & 7 \end{matrix}\right)$$ and basis $b = \{ (1,0,2), (0,...
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0answers
34 views

Basis of all real polynomials?

I am studying the book Topics in Algebraic Graph Theory by Beineke et all and the page 12. By the book, the set of all real polynomials can be generated by the set $\{1,x,x^2,\ldots\}$ which I ...
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1answer
44 views

Gröbner basis is not a vector basis?

We use the same notation for Gröbner basis and vector basis. I recall that $\langle 1\rangle_{GR}$ is the largest Gröbner basis while $\langle 1\rangle_{vector}$ is the smallest vector basis. So for ...
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2answers
77 views

Matrix equivalent to linear maps - sanity check

I'm reading some Linear algebra notes I found online, and am a bit confused about the following: If $U,V$ are finite dimensional $\mathbb{C}$-spaces with bases $(\mathbf{u}_1,\dots,\mathbf{u}_m)$ and ...
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1answer
36 views

Terminology: If $A, B$ are subspaces of $V$ and $A \cap B = \{0\}$ then they are …?

If $A, B$ are subspaces of $V$ and $A \cap B = \{0\}$ then ... If $V = A \oplus B$ they are complementary, otherwise I think that Halmos describes them as disjoint but this seems at odds with the ...
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1answer
78 views

Are the terms 'clan' and 'tribe' common in mathematics?

In the book 'Vector Measures' by Dinculeanu, he starts the discussion by talking about "classes of sets", and introduces two pieces of terminology I've never seen before, and can't find any evidence ...
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2answers
54 views

Understanding a proof of a theorem from S.Roman's “Advanced Linear Algebra”

There is a Theorem $1.5$ on page $43$ of the book "Advanced Linear Algebra" by Steven Roman. Theorem $1.5$. Let $F = \{ S_i | i \in I \}$ be a family of distinc subspaces of a vector space $V$. ...
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2answers
65 views

For a Vector Space $V = A \oplus B = A \oplus C \implies dim(B) = dim(C) $?

For a finite dimensional space there is no problem. $dim(V) = dim(A) + dim(B) = dim(A) + dim(C) \implies dim(B) = dim(C)$ For an infinite dimensional space it still holds that $dim(V) = dim(A) + ...
1
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1answer
43 views

Infinite matrix product

Let $$X=\left(\begin{array}{c} x_1 \\ x_2\\ \vdots \end{array}\right)$$ be an infinite real vector and $$A=(a_{ij}), \ 0<i,j<\infty$$ be an infinite real matrix. (1) For which $A$ can one ...
2
votes
2answers
32 views

angle between two planes, why can we use the dot product?

I understand that to find the angle we use the dot product of the normal vectors of the two planes, but why is it correct? as the normal vectors are both 90 degrees from the "real" angle of the planes
0
votes
0answers
26 views

minimum value of a directional derivative

$f=(x^2+y^2+z^2)e^{-(x^2+y^2+z^2)}$ find a point where the direction of the function as a minimum value and is parallel to the vector $3\hat{i}+2\hat{j}+\hat{k}$ So I took $\nabla f=(2xe^{-(x^2+y^2+...
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1answer
23 views

Showing a set is a root system in a vector space from definition of root system

Suppose I have the vectors $\alpha, \beta \in \mathbb{R}^2$ with inner products $(\alpha, \alpha) = 1$ and $(\beta, \beta) = 2$, and the angle between $\alpha$ and $\beta$ is $\theta = \frac{3\pi}{4}$....
3
votes
1answer
438 views

Proving any linear transformation can be represented as a matrix

I'm trying to prove that Theorem. Consider a linear transformation $T : \mathbb R^n \to \mathbb R^n$. The transformation $T$ can be represented as a matrix product $\mathbf x \mapsto A \mathbf x$...
24
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1answer
2k views

What is the logic/rationale behind the vector cross product?

I don't think I ever understood the rationale behind this. I get that the dot product $\mathbf{a} \cdot \mathbf{b} =\lVert \mathbf{a}\rVert \cdot\lVert \mathbf{b}\rVert \cos\theta$ is derived from ...
0
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1answer
27 views

vector generation by linear combination

I have 4 vectors in $R^3$ given as: $v1=(-1,2,0), v2=(3,1,2), v3=(4,-1,0), v4=(0,1,-1)$. I have to show that the vector $v= (5,6,0)$ can be generated by a linear combination of this vector. let the ...
0
votes
0answers
23 views

Directional derivative

what is the directional derivative of$ f(x,y)=xy+x^2$ at the point $(2,-1,1)$ in the direction $(1,3,-1)$? So the unit vector is $\frac{(1,3,-1)}{\sqrt{11}}$, now we have to take the gradient of ...