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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Find subspace $T$ of space $\mathbb R^3$ so that $\mathbb R^3=S \oplus T$

I have one problem. I am sure it is not complicated, but I only need help to see am I, at least, on the right path. Problem: Let $S=Span\{(0,-2,3),(1,1,1),(2, -2, 8)\}\subseteq \mathbb R^3$. Find ...
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1answer
29 views

Sylvester's argument for bilinear functions

Let $V$ be a vector space of dimension $n$ and let $b:\colon V \times V\to \mathbb{R}$ be a symmetric bilinear function. Sylvester's theorem says that there exists a basis of $V$ with respect to ...
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1answer
30 views

Basis for the space of linear transformations $L(\Bbb R^3,\Bbb R_3[x])$

How do I build a basis for the vector space $L(\Bbb R^3,\Bbb R_3[x])$? This is the vector space of all linear transformations that goes from $\Bbb R^3$ to the space of polynomials of degree 3 ...
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39 views

Wedge product is nondegenerate symmetric bilinear form

Let$$f: \Lambda^k(\mathbb{R}^n) \times \Lambda^{n - k}(\mathbb{R}^n) \to \mathbb{R}, \quad f(\alpha, \beta) = \alpha \wedge \beta.$$How do I see that $f$ is a nondegenerate symmetric bilinear form?
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48 views

Prove that $U$ is a vector-subspace

If $U$ is the set of all matrices that are commutative with the matrix $A$, show that $U$ is a vector subspace of the space $M^\mathbb{R}_{3\times 3}$ $$A=\begin{pmatrix}2&0&1\\ 0&...
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1answer
20 views

Problem with change of basis of an polynomial.

Good morning, i have a problem solving this: Express $a_{0}+a_{1}x+a_{2}x^{2}$ in terms of basis: $1,x-1,x^{2}-1$ I make this: $c_{1}1+c_{2}(x-1)+c_{3}(x^{2}-1)=c_{1}+c_{2}x-c_{2}+c_{3}x^{2}-c_{3}=...
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2answers
46 views

Does there exists an additive group homomorphism between two $K$-vector space that is not $K$-linear

My question is: Give me a field $K$. Can we always find two $K$-vector space $V_{1}$, $V_{2}$ and a map $f:V_{1}\rightarrow V_{2}$ such that: (1) If we view $V_{1}$, $V_{2}$ as additive group, then $...
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2answers
122 views

What does a norm of a polynomial space mean?

When talking about polynomial vector space, the following example was provided. A polynomial of degree $n$ in two variables is $$p(X)=\sum_{0\leq k+j \leq n} a_{j,k}x_1^jx_2^k$$ where $k+j=n$ and ...
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2answers
31 views

Prove that a union of bases for $S$ and $T$ is basis for $S + T$

Let $S$ and $T$ be subspaces of a vector space $V$. Assume that $B = \{ b_i | i \in I \}$ is a basis for $S \cap T$. Now, extend $B$ to a basis $A \cup B$ for $S$ where $A = \{ a_j | j \in J \}$ and $...
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0answers
19 views

Same vector space for arbitrary independent vectors?

If we use n linearly independent vectors x1,x2...xn to form a vector space V and use another set of n linearly independent vectors y1,y2...yn to form a vector space S, is it necessary that V and S are ...
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0answers
13 views

What are the Geometric Properties of Non Integer Vector Spaces?

I found a paper from Princeton called "Axiomatic Basis for Spaces with Non Integer Dimension" that presents five axioms and then starts to create a framework similar to what I'd think the subject ...
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0answers
34 views

The precise definition of Cartesian coordinate and Euclidean space?

I'd searched them for a while, but still have not found a clear and unity definition on it. The problem really confused me. What is the precise definition of Cartesian coordinate and Euclidean space? ...
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3answers
147 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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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
26 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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2answers
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 ...
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1answer
33 views

Notation: rotation matrix with a condition

I'm building a space simulation & am using this resource for converting Keplerian Orbit Elements to Cartesian Co-ordinates. The notation for step 6 has me slightly confused: Is the top part ...
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1answer
32 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
33 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
28 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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1answer
34 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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2answers
26 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 ...
0
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1answer
27 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 $W$, $...
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2answers
23 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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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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0answers
12 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
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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 ...
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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
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 ...
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33 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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28 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
44 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
52 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,\...
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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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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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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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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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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
37 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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2answers
56 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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1answer
46 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 ...
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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) + ...
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2answers
33 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
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0answers
33 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
29 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 ...
1
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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 $\theta = \frac{3\pi}{4}$....
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0answers
24 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 ...