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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1answer
42 views

Finite-dimensional subspace normed vector space is closed

I know that probably this question has already been answered, but I'd like to present my attempt of solution. Let $(E,\|\cdot\|)$ be a normed vector space and let $F\subseteq E$ be a ...
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0answers
21 views

vector subspace of all real polynomials which are divisible with $x^2 + 1$

Show that the set of all real polynomials which are divisible with $x^2 + 1$ is a vector subspace of space of all real polynomials to 4th degree. Also find base and dimension of this subspace. I ...
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2answers
38 views

Calculate the dimension of $U = \{(x_1,x_2,x_3,x_4,x_5) : x_1+x_3+x_5=x_2+x_4=0\}$

In the vector space $V \subset \Bbb R^5$, considering the vectors $v_1,v_2,v_3$ $v_1 = (0,1,1,0,0)$ $v_2 = (1,1,0,0,1)$ $v_3 = (1,0,1,0,1)$ We have $V = \mathrm{span}(v_1,v_2,v_3)$ ...
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2answers
60 views

Prove $\dim W \ge 2$

Let $U_1, U_2, W$ subspaces of a finite dimensional vector space, such that: $U_1 \cap U_2 = \{0\}$ $U_1 \cap W \ne \{0\}$ $U_2 \cap W \ne \{0\}$ Show that $\dim W \ge 2$. ...
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1answer
40 views

Subspaces of the set of real valued functions over an interval.

Show that the integral of all continuous real-valued functions on the interval [0,1] equal to b $\in$ R is a subspace of $R^{[0, 1]}$ if and only if b=0. So I am assuming that because both the ...
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1answer
31 views

$rk(A)=n$ implies $rk(AB)=rk(B)$

Let $A \in Mat_{m\times n}(\mathbb{R})$ and $B \in Mat_{n\times p}(\mathbb{R})$. Assume $rk(A)=n$. Prove that $rk(AB)=rk(B)$. Lets start by proving $rk(B) \ge rk(AB)$. Indeed, since the ...
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2answers
52 views

Show $L_1 \subseteq L_2$ or $L_2 \subseteq L_1$

Let $L_1,L_2$, two subspaces of a finite dimensional vector space. Prove that if $\dim(L_1 + L_2) = 1 + \dim (L_1 \cap L_2)$ then $L_1\subseteq L_2$ or $L_2 \subseteq L_1$. Well, I've read a ...
2
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1answer
25 views

Let $L,M$, two subspaces of $V$. Prove $L\cup M \ne V$ [duplicate]

Let $L,M$, two subspaces of $V$ where $L,M\ne V$. Prove $L\cup M \ne V$ My Try: Obliviously, $L\cup M \subseteq V$. It's left to show $L\cup M \subsetneq V$. $L,M$ have bases as subspaces. ...
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1answer
33 views

Using coordinates in B to find coordinates in B'

Part 1: $B=\{v_1,v_2,v_3\}$ is a basis of the real vector space $V$. ${B'}=\{v_1+v_2,av_1+v_3, bv_1-v_3\},\quad a,b\in\mathbb R$ What conditions should $a$ and $b$ satisfy for $B'$ to be a ...
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1answer
29 views

Understanding a definition for vector-spaces

Let $V$, a finite dimensional vector space, and $L$, a subspace of $V$. Let $T:V^*\rightarrow L^*$ defined as: $T(\varphi)(x)=\varphi(x)$ for all $\varphi \in V^*$. Prove $T$ is onto. Well, I'm ...
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3answers
29 views

Find vector and parametric equation of plane in $\Bbb R^{3}$ that passes through origin and is orthogonal to vector v.

$v=(4,0,-5)$ and I am given the hint: Construct two nonparallel vectors orthogonal to $v$ in $\Bbb R^3$. I've looked at this post Find the equation of the plane passing through a point and a vector ...
2
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1answer
20 views

Proof Concerning Linear Independence And Maximal Subsets

Serge Lang's Linear Algebra has, in chapter 1, a proof which seems rather long-winded. He wants to prove the following theorem: Theorem 3.1. let V be a vector space over the field K. Let ...
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1answer
41 views

How do I show that an endomorphism is self-adjoint if and only if $\langle u, Tu \rangle \in \mathbb{R}$ for all $u \in \mathbb{V}$

Let $$(V,\langle \cdot , \cdot \rangle)$$ be a complex vector space. Let $T \in \mathcal{L}(V)$ be an endomorphism. Now I want to show, that $T \in \mathcal{L}(V)$ is self-adjoint if and only if ...
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1answer
37 views

$T : M_{n \times n}(R) \rightarrow M_{n \times n}(R)$ and $T(A)= A^t$ and $ <A,B> = Tr(AB^t)$

Let $V = M_{n \times n}(R)$ with the inner product $ <A,B> = Tr(AB^t)$, and $T$ the linear operator given by $T : M_{n \times n}(R) \rightarrow M_{n \times n}(R)$ and $T(A)= A^t$ . How can i ...
2
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1answer
46 views

Isomorphism implies direct sum of Kernel and Image

If $f: U \rightarrow V$ and $g: V \rightarrow W$ are linear transformations between vector spaces over a field $K$ such that $ g \circ f$ is an isomorphism, then $V = \operatorname{Im}f \oplus ...
4
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1answer
50 views

Surjection of norms

Let $V$ be an infinite dimensional $\mathbb{C}$ (or $\mathbb{R}$) vector space. Suppose there exists two norms on $V$ such that \begin{equation*} \| \cdot\|_1 \leq \| \cdot \|_2. \end{equation*} Is ...
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1answer
49 views

Proof the that compact support is a vector space

Currently I am studying for my exam of Real Analysis, however there is one thing that I do not seem to get. Given: $$ \mathrm{Supp}(f):=\overline{\{x\in\mathbb{R}^n:f(x)\neq 0\}} $$ the support of ...
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2answers
28 views

If $f\in V$ of degree $n$ then for every $g \in P_n(\Bbb R)$ there exist scalars $c_0,c_1,..,c_n$ such that $g = c_0f + c_1f'+ … + c_nf^{(n)}$

Let $V=P(\Bbb R)$ and $1 ≤ i$ be the vector space of the polynomials with real coefficients, on the field of real numbers $\Bbb R$. Let $T_i(f)=f^{(i)}$ the $i$th derivate of $f$. a) I have to show ...
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4answers
83 views

Sums of solutions to $z^n-1 = 0$ that equal 0

Consider the solutions of the equation $z^n - 1 = 0$, where $z$ is a complex number: ${z_1,z_2...z_n}$. What are ALL the possible sums $\sum_{i=1}^n a_iz_i$ over these n solutions, where $a_i$ are ...
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4answers
83 views

$\{ v_1,v_2,…,v_n\}$ is basis of $V$ if and only if $\{ v_1,v_1 + v_2,…,v_1 + v_2+…+v_n,\}$ is a basis of $V$

Let $V$ a vector space over a field $K$. Is it true $\{ v_1,v_2,...,v_n\}$ is basis of $V$ if and only if $\{ v_1,v_1 + v_2,...,v_1 + v_2+...+v_n,\}$ is a basis of $V$ ? I made some examples and ...
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2answers
30 views

Show the following subspaces are invariant

Let $V$ be a vector space over a field $F$ and let $\alpha \in End(V)$. IF $W$ and $Y$ are subspaces of $V$ which are invariant under $\alpha$, show that both $W+Y$ and $W\cap Y$ are invariant under ...
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3answers
55 views

Considering Vectors Geometrically

I have a few questions which a little research (searching the internet through Google) has not satiated. It seems that vectors are very important, even when considering them as the arrows which ...
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1answer
13 views

$x\in{\rm span}(S\cup\{y\}), x\notin{\rm span}(S)$ implies $y\in{\rm span}(S\cup\{x\})$

If $x\in{\rm span}(S\cup\{y\})$ and $x\notin{\rm span}(S)$, then $y\in{\rm span}(S\cup\{x\})$ This statement is simple enough if ${\rm span}$ is defined in terms of finite linear combinations: if ...
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1answer
18 views

Vectors, columns and representations

When I learned quantum mechanics, my professor frequently emphasized that a matrix is a representation of an operator, not the operator itself, and a column ($1\times M$ matrix) is not a vector, it's ...
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1answer
33 views

Show that $v_1, v_2, \cdots , v_r$ span $R^n$ if and only if $v_1+\alpha *v_2, v_2, \cdots, v_r$ span $R^n$

In our textbook I've seen this claim: $v_1, v_2, \cdots , v_r$ span $R^n$ if and only if $v_1+\alpha *v_2, v_2, \cdots, v_r$ span $R^n$ for some $\alpha \in \mathbb{R}$ Now I've tried to prove it to ...
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1answer
29 views

Show that $\DeclareMathOperator{im}{Im} \im(\alpha) \cap \im(\beta)={0_v}=\ker(\alpha) \cap \ker(\beta)$

I am just completely stuck on this problem, it may just be me confusing the vocabulary and what they mean Question: Let V be a finite-dimensional vector space over a field F and let $\alpha, \beta$ ...
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2answers
33 views

Showing $Y'$ is isomorphic to $Y''$

Let $V$ be a vector space over a field $F$ and let $W$ and $Y$ be subspaces of $V$ satisfying $W+Y=V$. Let $Y'$ be a complement of $Y$ in $V$ and let $Y''$ be a complement of $W\cap Y$ in $W$. Show ...
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2answers
62 views

Show ker($\alpha$)=ker($\alpha$)^2 iff ker($\alpha$) and im($\alpha$) are disjoint

Let $V$ be a vector space over a field $F$ and let $\alpha$ be an element of $\operatorname{End}(V)$. Show $\ker(\alpha)=ker(\alpha^2)$ iff $\ker(\alpha)$ and $\operatorname{im}(\alpha)$ are disjoint. ...
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1answer
46 views

A question about vector fields and divergence

I am reading the paper http://www.goshen.edu/physix/mathphys/gco/TensorGuideAJP.pdf in order to gain a basic understanding about tensors. I had some difficulties about understanding some definitions. ...
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1answer
35 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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2answers
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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25 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
70 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
23 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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30 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 ...
1
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2answers
32 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 $(x-y)\perp u \ \ : \ \ \forall u \in W$ ...
2
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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 ...