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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2answers
540 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 ...
0
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2answers
22 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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0answers
16 views

Prove that then the linear mapping is surjective

Let in the field $F$: $2 \ne 0$. Let $Q$ is a nondegenerate quadratic form on a finite-dimensional vector space $V$. Suppose that $Q(v) = 0$ for some nonzero vector $v \in V$. Prove that then the ...
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1answer
21 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 ...
-1
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0answers
7 views

How can I find the necessary speed and speed of rotation for a problem from a parametric equation?

I have been given the following questions for a project that I am currently working on: Questions 1 to 8 I have completed questions 1 through 6 but have no idea how to do questions 6 or 7 after ...
0
votes
1answer
18 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
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
25 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
21 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? ...
1
vote
1answer
55 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
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1answer
47 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: ...
15
votes
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 ...
1
vote
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$ ...
7
votes
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 ...
2
votes
1answer
593 views

Triangle and parametric coordinates

I'm studying on a book where it says: "A triangle is the set of points where for some point po, where u and v range over the parametric coordinates (we are talking about barycentric coordinates ...
1
vote
0answers
10 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 ...
2
votes
1answer
15 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
votes
0answers
38 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
vote
2answers
21 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
22 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 ...
0
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2answers
22 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 ...
2
votes
2answers
504 views

Space spanned by matrices

I have a set of $5$ by $5 $matrices, $M_1,M_2,...,M_{19} ,M_{20}$. I want to try to find a basis from this set and also to find relationships between these matrices. This is how I think I should ...
0
votes
0answers
42 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?
2
votes
1answer
39 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 ...
3
votes
1answer
203 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
56 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 ...
0
votes
0answers
25 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
votes
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 ...
1
vote
1answer
43 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 ...
4
votes
2answers
72 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 ...
0
votes
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 ...
5
votes
1answer
73 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 ...
0
votes
2answers
48 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 $V$. ...
4
votes
2answers
64 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
vote
1answer
37 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 ...
1
vote
2answers
441 views

Angle between two vectors not in same plane

I want to know how calculate the angle between two vectors and both are not in same plane, which means that they don't intersect at any point? For example how do I calculate angle between AB and EF ...
2
votes
2answers
27 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 ...
3
votes
1answer
5k views

How to find perpendicular distance from point to plane in $3D$.

The line $L_1$ passes through point $A$ whose position vector is $3i - 5j + 4k$, and is parallel to the vector $3i + 4j + 2k$. The line $L_2$ passes through the point $B$ whose position vector is $2i ...
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 ...
1
vote
1answer
22 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 = ...
3
votes
1answer
388 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 ...
24
votes
1answer
1k 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
votes
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 ...
-1
votes
0answers
11 views

the direction where the directional derivative is maximal

I know that to find the directional derivative we use: $\nabla f\cdot \hat{u}$ where $\hat{u}$ is the direction where we seek the derivative. we know that the directional derivative is maximal in the ...
0
votes
0answers
21 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 ...
1
vote
0answers
18 views

Represent of multilinear map [duplicate]

Let $V_1,V_2$ be vector space and $\{e_i\},\{\overline e_i\}$ are basis respectively. $\forall ~l\in L(V_1,V_2; F)$ ,why $l$ can be represented as $$ l=\sum\limits_{ij} a_{ij} \omega^i\otimes ...
0
votes
1answer
20 views

Prove that a specific subset $A$ of a nontrivial vector space $V$ over an infinite field $\mathbb{F}$ is infinite

Let $V$ be a nontrivial vector space over an infinite field $\mathbb{F}$. Suppose $V = \bigcup\limits_{i=1}^{n} S_i$, where $S_i$ is a proper subspace of $V$. We assume that $S_1$ is not included in ...
1
vote
1answer
156 views

Orthogonal projection question

Consider the (orthogonal) projection $T: \mathbb{R}^3 \to \mathbb{R}^3$ onto the plane $x - y + z = 0$. (a) Find the standard matrix $[T]_S$ for $T$. (b) Find a new basis $B$ so that ...
1
vote
1answer
19 views

Symmetric, Antisymmetric, and Alternating Bilinearforms form a vector subspace

I have to show that the space of symmetric, the antisymmetric and the alternating bilinear forms each form a vector subspace of the space of all bilinear forms $\operatorname{Bil}(V,K)$ with $V$ being ...
3
votes
1answer
123 views

How to find the appropriate weights to maximize the third coordinate while the first two are zeros

Let's assume, that $v_1, ..., v_n \in \mathbb{R}^3 $ and $ \lambda_1, ..., \lambda_n \in [0, 1] $ The $ v_1, ..., v_n $ vectors are given. I have to find the appropriate weights ($ \lambda_1, ..., ...