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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36 views

finding the dimension and a basis of U given a single vector

I've been given a very confusing homework problem that is as follows: Let U be the set of all vectors u in $ℝ^4$ such that $2(u_1) + 3(u_3) - 2(u_4) = 0$ (i.e. U is the solution space of a given ...
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0answers
24 views

Notation for a vector space: $(\mathbb{C}^\infty)^{\otimes L}$

In a paper, the authors use the notation $(\mathbb{C}^\infty)^{\otimes L}$, where $L$ is a constant, for a vector space, but they do not give a definition. They also implicitly introduce an inner ...
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2answers
48 views

Weird field notation

I have a question: Let $\mathbb{F}$ be any field characteristic $0$. Recall that $x_i$, denotes the $i^{th}$ entry of a vector $x\in\mathbb{F}^n$. Define $$S = \{x\in\mathbb{F}^5 \mid x_i = ...
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44 views

$\dim(V) = \dim T(V) + \dim T^{-1}(0)$

Let $T\colon V \rightarrow W$ a linear transformation between the real vector spaces $V$ and $W$ both with finite dimension. How can i prove that $\dim(V) = \dim T(V) + \dim T^{-1}(0)$. I can't ...
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1answer
27 views

Given two linear transformations, prove that V is equal to the direct sum of the kernels.

Let $S:V\to V$ and $T:V\to V$ be two linear transformations such that: $T^{2} = S^{2} = 0, T\circ S + S \circ T = Id$. Prove that $V= Ker(S)\oplus Ker(T)$. What can I use to prove this? Honestly, ...
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26 views

Matix column-wise multiplication operator

I'm trying to find the proper operator for a column wise multiplication. Consider $v=[v_1, v_2, ..., v_n]^T$ and $A=\begin{bmatrix} a_{1,1} & a_{1,2} & a_{1,3} \\a_{2,1} & a_{2,2} & ...
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35 views

Write down a matrix of which only the null space is known?

What is the matrix in which null space are all of the multiples of the vector: $$\vec{v}=\begin{bmatrix}4 \\ 3 \\ 2 \\ 1\end{bmatrix}$$ I suppose there are a lot of solutions, but I don't I am not ...
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3answers
93 views

How can a subspace have a lower dimension than its parent space?

If $V$ is a vector subspace of $W$, then $$\dim(V) \le \dim(W)$$ Why? Does that mean that for $$W = \mathbb{R}^3\\ V = \{(0,0)\}$$ $V$ is a valid subspace of $W$? But $V$ only has two ...
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3answers
71 views

Understanding coordinates with respect to orthonormal bases.

If you have an orthonormal base $B$ of $\mathbb{R}^2$, you can calculate coordinates $\vec{x} \in \mathbb{R}^2$ with respect to $B$: $$[\vec{x}]_B = (\vec{x}\cdot B_1 , \vec{x}\cdot B_2)$$ I know ...
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1answer
25 views

Orthogonal complement of a line in $\mathbb{R}^2$ spanned by $(1,2)$

Have a subspace $W$ of $\mathbb{R}^2$, where $W$ is spanned by $\{(1,2)\}$. Determine $W^\perp$. Well then, clearly $W$ is a line in a 2D space. So I guess $W^\perp$ is a line, too. $W^\perp$ ...
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1answer
35 views

Few basic things unclear to me about inner product spaces and orthonormal basis

Few things unclear to me about inner product spaces: assume V is an inner product space with B orthonormal basis. Why is it true that: $$\langle x,y\rangle = \langle[x]_{B} , [y]_B \rangle{st}$$ ...
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1answer
29 views

Bases of subspaces.

I don't understand how can we prove this. Find a basis of the following subspaces of $\mathbb{R}^4$: a. The vectors $x = (x_1, x_2, x_3, x_4)$ where $x_1=2x_4$ b. The vectors for which $x_1 + x_2 + ...
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2answers
31 views

Going from Linear algebra to Multivariable Calculus [closed]

I just finished a course in Linear algebra, can anyone tell me how Linear and multivariable calc are related?
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1answer
25 views

About vector orthogonality with itself and implication in a subspace's complement.

My definition of vector orthogonality is simply that they are if their dot product is $0$. I saw a definition that says The orthogonal complement of a subspace in $\mathbb{R}^n$is the set of ...
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3answers
64 views

Determine the equation of a line that passes through point $A(1,0,2)$ and intersects the line $r=(-2,3,4) +s(1,1,2)$, at a right angle.

Vector/Linear algebra question. I already have the solutions manual but I still don't understand how to arrive at the answer. Would appreciate some help, thank you. $r$ is a vector equation for a line ...
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1answer
61 views

Dynamics of matrices over finite field and Similarity of matrices

Consider a set $M$ of all possible square matrices over a finite field $F_p$. Now consider a map $f_A(x)=A.x$ where $x$ $\in$ $M$ and also the matrix $A$ is a member of $M$. It is needless to mention ...
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2answers
33 views

Quotient spaces and quotient groups: equivalence classes and cosets

(Throughout this post, I am talking about vector spaces.) I had the pleasure of doing Abstract Algebra two semesters early, however, I feel like some general context was lost in the process. While I ...
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0answers
35 views

Faster way of finding critical points?

So I am looking at parametric vector function. $$ \begin{vmatrix} \cos (t) & -\sin (t) & 0 \\ \cos f(t) \sin (t) & \cos f(t) \cos (t) & -\sin f(t) \\ ...
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2answers
26 views

$T: V \rightarrow V$ a linear transformation such that $T^2 = I$ and $H_1= \{v \in V | T(v) = v\}\ $ and $H_2= \{v \in V|T(v) = -v\}\ $

Let V a vector space and $T: V \rightarrow V$ a linear transformation such that $T^2 = I$ and $H_1= \{v \in V | T(v) = v\}\ $ and $H_2= \{v \in V|T(v) = -v\}\ $ then $V = H_1 \bigoplus H_2$ I stuck ...
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1answer
23 views

If a vector fulfils one condition in this theorem, does it automatically fulfil both?

I have this theorem: If $W$ is a subspace of $\mathbb{R}^n$, for any $x\in \mathbb{R}^n$ there will exist some unique $y\in W$ such that $(x-y)\perp u \ \ : \ \ \forall u \in W$ ...
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1answer
26 views

Understanding the orthogonal complement of a subspace.

This is my definition of orthogonal complement: Given a vector subspace if $\mathbb{R}^n$, its orthogonal complement is the set of all vectors in $\mathbb{R}^n$ that are orthogonal to any ...
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2answers
88 views

maximum and minimum dimension of the space generated by $\{v_1,v_2,v_3,v_4\}$

I'm confused about this problem. I have four vectors $v_1 = (1,1,1,a), v_2 = (1,2,3,a), v_3= (b,1,0,1), v_4 = (0,b,0,0)$ with $a,b$ real numbers. Determine the maximum and minimum dimension of the ...
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1answer
33 views

Find the signature of $Q(x_1,\ldots,x_n)= \sum_{i,j=1}^{n} a_ia_jx_ix_j$

In $\mathbb{R}^n$ let $Q(x_1,\ldots,x_n)= \sum_{i,j=1}^{n} a_ia_jx_ix_j$ quadratic form. $a:=(a_1,\ldots,a_n)\neq0$ $\in \mathbb{R}^n$ find the signature of $Q$
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0answers
25 views

About an orthogonal complement theorem

Let $W$ be a subspace of $\mathbb{R}^n$. For any vector $x \in \mathbb{R}^n$, there will one unique vector $y \in W$ that fulfils: $$(x-y) \perp w \ \ : \ \ \forall w \in W$$ I have trouble ...
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2answers
33 views

Determining that this basis is linearly independent with a variable

Have the basis $$B = \{ (1,2,0) , (1,1,1) , (1,a,0) , (0,0,a) \}$$ Explain why doesn't this basis have a dimension of $4$. The only way would be, I guess, that it is linearly dependent, ...
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1answer
31 views

Coordinate vector of a subspace of $\mathbb{M}_{2,2}(\mathbb{R})$

Have $$\left\{ \left( \begin{matrix} x & y \\ y & x + y \end{matrix} \right) : x,y,\in \mathbb{R}\right \}$$ Which is a vector subspace of $\mathbb{M}_{2,2}(\mathbb{R})$. I was asked ...
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1answer
36 views

Left shift operator $L: l^2 \rightarrow l^2$ on the sequence space $l^2$

$$L: l^2 \rightarrow l^2$$ is defined by $$b = (b_1,b_2,...) \mapsto Lb = (b_2,b_3,...)$$. $(Lb)_n = b_{n+1}$ respectively. How can I determine the adjoint endomorphism $L^*$? Kind regards George
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2answers
22 views

restricting invertible maps to get new maps

For V and W as vector spaces, let we define V ⊗ W and suppose T be a invertible linear map from V ⊗ W to itself with special condition, I want to know whether there exist something like restricted ...
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2answers
103 views

curl of (cross product of two vectors), i know the formula, but not sure how to prove it

$$\text{curl } \left(\textbf{F}\times \textbf{G}\right) = \textbf{F}\text{ div}\textbf{ G}- \textbf{G}\text{ div}\textbf{ F}+ \left(\textbf{G}\cdot \nabla \right)\textbf{F}- \left(\textbf{F}\cdot ...
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4answers
89 views

Geometrically, what is the span of vectors?

Simple question from a calc 3 beginner. Visually I cannot imagine the span of two vectors, what does this necessarily mean? For example my text mentions if two vectors are parallel their span is a ...
1
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1answer
34 views

Is $\mathcal{L}^p \subset \mathcal{L}^{p-1} $?

A random variable $X$ is called integrable if $E[X] < \infty$. We say that $X \in \mathcal{L}^1$ if $E[X] < \infty$, and in general $X \in \mathcal{L}^p$ if $E[|X|^p] < \infty$. I know that ...
0
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1answer
26 views

Linear independent vectors of nilpotent transformation

$V$ is a vector space. $N$ is a nilpotent transformation $N:V\rightarrow V$ such that $N^k=0$ ($k$ is the lowest). $v \in V$, $v \notin \text{ker}\ N^{k-1}$ (in other words: $N^{k-1}v \ne 0$). Let ...
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1answer
12 views

Given two sets of vectors, is there a relationship that describes whether one of them is “orthogonal” to another?

We saw this theorem regarding orthogonal vector subspaces: Have $$A = \{a_1,a_2,a_3,...,a_k\}\\ B = \{b_1,b_2,b_3,...,b_r\}$$ Bases of vector subspaces $S$ and $T$ respectively. Then: ...
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1answer
22 views

What does the $t$ in $(x,y,z)^t$ mean?

Just a question on notation. I have seen a plane defined this way: $$S = \{(x,y,z)^t \in \mathbb{R}^3 \ / \ 2x-3y+z = 0\}$$ See the $t$ superscript on $(x,y,z)$? Well, I am not quite sure what is ...
2
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1answer
41 views

2x2 symmetric matrix is a subspace of vector space.

Can you kindly check my proof of the problem and correct if possible. The following $S=\{A\in M_{2,2} | AA^T=A^TA\}$ is a subspace of $V=M_{2,2}$ all real $2\times2$ matrices. My proof: S ...
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0answers
23 views

q-analog of vector space dimension

I am reading about "quantum dimension" $\dim_q V$ where $V$ is a vector space. In fact, you could write it $[\dim V]_q$ where $\dim V$ is the dimension of the vector space and $[n]_q = ...
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3answers
19 views

How to find the vector that passes through a point and is perpendicular to another vector.

Let $ \mathbb{a} = i+4j-3k$ and $b = 7i+20j-12k$ be vectors and $A(2,5,-3)$ be a point. I want find the line $l_ 3$ passing through point $A$ wich is perpendicular to both veotors. How should I do ...
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1answer
14 views

Eigenvector shared by two endomorhisms

I am guessing if the following fact is true: Let be $V$ a finite vector space above a field $K$. Let $f, g$ be two endomorphisms of $V$ with $f g = g f$. We assume that both $f$ and $g$ have got at ...
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1answer
23 views

Endomorphisms and Invariant Subspaces

I have a question or two regarding the following exercise: Let $\alpha$ be the endomorphism of $\Bbb{Q}^4$ defined by: $$\alpha : \left[\begin{matrix}a \\ b \\ c \\ d \end{matrix}\right] \mapsto ...
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0answers
19 views

Existence of linear/affine subspace for a number of vectors

Let $V$ be a vector space over a field $K$. Let $k \le \dim V$ be a natural number. I want to show that for each k vectors $v_1, ..., v_k$ there is a linear subspace $U$ of $V$ which has dimension ...
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3answers
39 views

Why does linearly independent spanning set imply minimal spanning set for a vector space?

Suppose β is a linearly independent spanning set of some vector space V. Why must it be the minimal spanning set? In other words, why can there not be two linearly independent spanning sets of a ...
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2answers
21 views

P3 being subspace of vector space?

V = P3 (all real polynomials of degree at most 3) and $S = \{p(x)\in P_3 | x·p'(x) = p(x),\} $ is it a subspace of vector space $V$? Solution: I don't even know is it possible for the equation ...
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2answers
52 views

Difference between Euclidean space and vector space?

I often hear them used interchangeably ... they are very complicated to make any use of. Wikipedia words: Euclidean space: One way to think of the Euclidean plane is as a set of points ...
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1answer
20 views

fields and subspaces

Let F be a field and let V=F^F, which is a vector space over F. Let w be the set of all functions f element of V satisfying f(1)=f(-1). Is W a subspace of V? a. Has the zero vector b. closed under ...
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1answer
60 views

Show that Z cannot be turned into a vector space over any field. [duplicate]

Show that Z cannot be turned into a vector space over any field. So, we have 2 cases here. Case 1:lets suppose the charF=P, n does not equal 0, then (1+1+...+1)n=1n+1n+...+1n=n+n+...+n=pn=wchich ...
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1answer
46 views

Finding loci of possible points satisfying vector simultaneous equations

I recently had an exam and a question came up which I was only partially able to answer. The question was the following: Let $\vec{a}$, $\vec{b}$ and $\vec{c}$ be constant vectors in ...
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2answers
18 views

dimension of the vector space using matrices

Let $C$ be an $n \times n$ real matrix. Let $W$ be the vector space spanned by $\{I, C, C^2, \ldots C^{2n}\}$. The dimension of the vector space $W$ is $ 1.\ 2n \hspace{4cm} 2.\ \text{at ...
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1answer
32 views

problems with finding a basis

Given is : $\mathbb{R^\mathbb{R}_f}:=\{ \alpha:\mathbb{R} \longrightarrow \mathbb{R}| \alpha(x)=0, \}, \alpha(x)\ne0$ only at finitely many points.Show that: $\mathbb{R^\mathbb{R}_f}$ is a subspace ...
0
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1answer
52 views

difficulties with prooving: K is a vector space over Z/pZ

I am trying to solve the followong exercise: Given is K as a field with finitely many elements. i) show that K is a vector space over $\mathbb{F}_p:=\mathbb{Z/p\mathbb{Z}}$, for some special values ...
3
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1answer
23 views

Prove is linearly independent

Prove that that the following subset $S \subseteq V$ in the respectively specified $K$- vector space $V$ is linearly independent a. $K=R$, $ V=R[x] $, $S$= {$x^n-x^m| n,m ∈ R,$ n-even, m-odd}