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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168 views

Show that $F+G$ is closed when $G$ a closed subspace of normed space $E$ and $F$ a finite dimensional subspace of $E$.

Question: Let $E$ be a normed space. Let $G$ be a closed subspace of $E$ and let $F$ be a finite dimensional subspace of $E$. Show that $F+G$ is a subspace of $E$ and is closed. I'm having trouble in ...
4
votes
1answer
196 views

Normal $T\in B(H)$ has a nontrivial invariant subspace

I am wondering if the following is true: Every normal $T\in B(H)$ has a nontrivial invariant subspace if $\dim(H)>1$?
4
votes
1answer
386 views

Algorithm to find the basis of intersection of subspaces without gaussian elimination.

Is there an algorithm to find the basis of intersection of subspaces $A_1$ and $A_2$, if we have the bases of subspaces $A_1$ and $A_2$, without using Gaussian elimination? Thanks.
4
votes
1answer
500 views

When are two diagonal matrices congruent?

This is probably a question that does not admit a simple answer. However, I'd like to know whether there exist criteria that determine when two diagonal matrices are congruent. I have the suspicion ...
4
votes
3answers
490 views

Does $\det(A) \neq 0$ (where A is the coefficient matrix) $\rightarrow$ a basis in vector spaces other than $R^{n}$?

I know that for a set of vectors $\{ v_{1}, v_{2}, \ldots , v_{n} \} \in \mathbb{R}^{n}$ we can show that the vectors form a basis in $\mathbb{R}^{n}$ if we show that the coefficient matrix $A$ has ...
4
votes
1answer
354 views

Prove a basic fact on a linear combination of vectors

Let $x_i (i=1,...,n, n>d)$ be a unit vector in $R^d$. $c_i>0$ is a positive real scalar. How to prove the following fact? Fact: There exist some vectors $x_i$ such that $\sum_{i=1}^n c_i ...
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2answers
251 views

Dot product of two vectors

How does one show that the dot product of two vectors is A · B = |A| * |B| * cos(Θ) ?
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2answers
477 views

Is $(\mathbf{V} \cap \mathbf{W})^{\bot}=(\mathbf{V}^{\bot} \cap \mathbf{W}^{\bot})$?

Is $(\mathbf{V} \cap \mathbf{W})^{\bot}=(\mathbf{V}^{\bot} \cap \mathbf{W}^{\bot})$? I tried element-chasing, but I am getting confused when trying to determine mutual containment.
3
votes
2answers
68 views

Prove that, if $\{u,v,w\}$ is a basis for a vector space $V$, then so is $\{u+v, v+w, u+v+w\}$.

I'm trying to prove the following statement: In a vector space $V$ over a field $\mathbb{F}$, if $\{u,v,w\}$ is a basis for $V$, then $\{u+v, u+v+w, v+w\}$ is also a basis. $$\underline{\text{My ...
3
votes
1answer
107 views

How many (unordered) bases does $\Bbb F_q^n$ have as a vector space over $\Bbb F_q$?

Following the recommendation here to get this question out of the unanswered queue, I've changed this from a proof-verification question into an answer-your-own. Here's the question again in case ...
3
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4answers
276 views

If $X$ is an orthogonal matrix, why does $X^TX = I$?

It's not immediately clear to me why this is true. My notes say that putting $n$ orthonormal vectors $ v_1, ..., v_n$ in the columns of $X$ gives $X^TX = I$, and it follows from this that the rows of ...
3
votes
2answers
2k views

Relation between Interior Product, Inner Product, Exterior Product, Outer Product..

Following my previous question Relation between cross-product and outer product where I learnt that the Exterior Product generalises the Cross Product whereas the Inner Product generalises the Dot ...
3
votes
1answer
174 views

Define two differents vector space structures over a field on an abelian group

Exercise 3 from Roman's book "Advanced Linear Algebra". The author asks us to "find an abelian group $V$ and a field $\mathbb{F}$ for which $V$ is a vector space over $\mathbb{F}$ in at least two ...
3
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1answer
677 views

How many different basis' exist for an n-dimensional vector space in mod 2?

Imagine a 2-dimensional vector space in $\mathbb{Z} /2 \mathbb{Z}$. The only possible basis' are $$ \left(\begin{array}{c} 1\\ 0 \end{array}\right), \left(\begin{array}{c} 0\\ 1 ...
3
votes
2answers
227 views

How to prove the inequality $\Theta(x,y)\le \Theta(x,z)+\Theta(z,y)$?

Let $x, y$ be two complex vectors, $$\cos\Theta(x,y):=\operatorname{Re} \frac{y^*x}{\|x\|\|y\|} .$$ Then I want to prove that $$\Theta(x,y)\le \Theta(x,z)+\Theta(z,y) .$$
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votes
2answers
41 views

How to figure the size of the following vector set?

Let $V$ be the set of all vectors in $\mathbb R^n$ with entries $±1$. What is the size of this vector set? I know the answer is $2^n$ but I cannot prove why. I feel like this has something to do ...
2
votes
2answers
553 views

Linear Algebra, Vector Space: how to find intersection of two subspaces ?

$${ W = Sp\{{(1,3,4),(2,5,1)\}}\\ U = Sp\{{(1,1,2),(2,2,1)}} \}$$ Find a span $${U \bigcap W}$$ First time using Math latex, pretty hard.
2
votes
1answer
109 views

Linear Transformation defined by a Matrix and Invariant Subspaces

I got stuck solving this problem: Let $T:\mathbb{R}^3\to \mathbb{R}^3$ be the linear transformation defined by the matrix A in the standard basis of $\mathbb{R}^3$, $E=\{e_1,e_2,e_3\}$ ...
2
votes
4answers
347 views

Find a vector that is perpendicular to $u = (9,2)$

Attempt: We know perpendicular vectors have dot product $u \cdot v = 0$ therefore $[9,2] \cdot [x,y]$ = 0 $9x + 2y = 0$ what would I do now? thanks!
2
votes
4answers
186 views

Intuition behind definition of transpose map

The linear map $T^t:V' \to V'$ (where $V'$ is the set of all linear maps from $V$ to its scalar field $\Bbb F$) is defined by : $$T^t(f)(v)=f(T(v))$$ This looks like some "commutative" definition ...
2
votes
1answer
438 views

Calculate angle of triangle

I need to calculate the angle between two sides, I have the length of A & B sides, but don't know how to find the angle... Both sides are the same length. I can get the start and end vectors of ...
2
votes
1answer
1k views

Relationship between covariant/contravariant basis vectors

I'm starting to learn some of the basics of covariant and contravariant vectors. I'm a little confused about the difference between a covariant and a contravariant basis vector. I know that the ...
2
votes
2answers
94 views

How to show $X=\{A\in\mathcal{L}(\mathbb R^m, \mathbb R^n); \textrm{Ker}(A)=\{0\}\}$ is open in $\mathcal{L}(\mathbb R^m, \mathbb R^n)$?

How to show $X=\{A\in\mathcal{L}(\mathbb R^m, \mathbb R^n); \textrm{Ker}(A)=\{0\}\}$ is open in $\mathcal{L}(\mathbb R^m, \mathbb R^n)$? Here $\mathcal{L}(\mathbb R^m, \mathbb R^n)$ is the set of ...
2
votes
4answers
93 views

Should I use sets or tuples when dealing with linear dependence?

Let set of vectors $\{x,y,z\}$ be linearly independent. Then would $\{x,y,z,x\}=\{x,y,z\}$ be linearly dependent, also? If so, that seems like a problem (since $\alpha x+\beta y+\gamma ...
2
votes
1answer
173 views

Counting automorphisms

How does one count the number of automorphisms of a vector space? If a vector space over $\mathbb F_p$ has $n$ ordered bases how many are there? I think I should be considering the mappings of a set ...
2
votes
0answers
215 views

Direct sum and direct product of vector spaces [duplicate]

Possible Duplicate: The direct sum $\oplus$ versus the cartesian product $\times$ (Definition) I was wondering how their definitions are different? Are they both the cartesian product with ...
2
votes
3answers
2k views

Geometric interpretation of the multiplication of complex numbers?

I've always been taught that one way to look at complex numbers is as a cartesian space, where the "real" part is the x component and the "imaginary" part is the y component. In this sense, these ...
1
vote
1answer
23 views

Meaning of passing from a column to a row vector

When passing from column to row vectors in $K^n$ conceptually we're passing from a vector $(a_1,\ldots , a_n) \in K^n$ to it's associated linear functional defined by $ f(x_1,\ldots , x_n)=\sum_i a_i ...
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1answer
68 views

Vector Spaces: Tensor Product

Reference Foundation for: Hilbert Spaces: Tensor Product Problem Given a vector spaces $V$ and $W$. Take its algebraic tensor product: $\tau:V\times W\to V\otimes W$ How to prove that the image ...
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3answers
82 views

Prove that $\mathrm{span}\{ I,A,A^2… \} = \mathrm{span} \{ I,A,A^2,…, A^{k-1}\}$

Let $A\in M_n(F)$ and $k=\deg(m_A)$ where $m_A$ is the minimal polynomial of $A$. Prove that $\mathrm{span}\{ I,A,A^2... \} = \mathrm{span} \{ I,A,A^2,..., A^{k-1}\}$ So we have that $m_A = a_0 ...
1
vote
1answer
20 views

Spans and Dot Product: Findin the linear combination

Suppose $(v_1, v_2, v_3)$ is a set of vectors mutually perpendicular. Assume that $\|v_1\|= \sqrt{27}\quad \|v_2\| = \sqrt{14}\quad \|v_3\|= \sqrt{ 4}\ $ Let $w$ be a vector in ...
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vote
0answers
55 views

Can a Norm be Induced by two Different Complex Inner Products?

Let $(X,\|\cdot\|)$ be a normed vector space over $\mathbb{C}$. If $\|x\|=\sqrt{\langle x,x\rangle}$ and $\|x\|=\sqrt{\langle x,x\rangle'}$ for all $x\in X$ where $\langle,\rangle$ and ...
1
vote
1answer
42 views

$U_1\oplus W=V$ and $U_2\oplus W=V$ but $U_1 \neq U_2$ where $U_1$ and $U_2$ are two subspaces of $V$.

I am searching some counterexamples such that $U_1\oplus W=V$ and $U_2\oplus W=V$ but $U_1 \neq U_2$ where $U_1$ and $U_2$ are two subspaces of $V$ and $V$ is a vector space except $\mathbb {R}^2 ...
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vote
2answers
107 views

Find a matrix such that $Ax=0$

Let $$W = span\left\{ {\left( {\matrix{ 1 \cr 0 \cr 0 \cr 1 \cr } } \right),\left( {\matrix{ 0 \cr 2 \cr 1 \cr { - 1} \cr } } \right)} \right\}$$ I was asked ...
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vote
3answers
71 views

Understanding dot product of continuous functions

I'm reading about Fourier analysis and in my book the author speaks about dot product for continuous functions $f, g\in L^2(a,b)$(the set of functions which are square-integrable on the interval ...
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vote
2answers
46 views

Is $S$ a subspace of $V$?

Let $V$ be the set of real-valued continuous functions on the interval $[-3, 3]$. $S$ is set of real-valued functions with condition $f(-1) = f(1)$. Is $S$ a subspace of $V$? Prove, and if not, why?
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vote
4answers
108 views

Basis on vector space $V$

If $S_i$ is a set of linearly independent vectors of vector space $V$ and $S_g$ a set of generators of $V$. Prove that it exist $S'_g\subset S_g$ that $S_i\cup S'_g$ is a basis of $V$. Notice that ...
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vote
2answers
52 views

the rank of a linear transformation

Let $V$ be vector space consisting of all continuous real-valued functions defined on the closed interval $[0,1]$ (over the field of real numbers) and $T$ be linear transformation from $V$ to $V$ ...
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2answers
114 views

Question regarding sub-spaces of finitely generated vector space

Let $L_1,L_2$ be sub-spaces of finitely generated vector space. Prove that if $\dim(L_1+L_2)=1+\dim(L_1 \cap L_2)$, then $L_1 \subseteq L_2$ or $L_2 \subseteq L_1$. Unfortunately, I don't ...
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vote
1answer
337 views

Problem with alternate solution — Equation of plane through point and containing intersection line of two planes [Stewart P $803, 12.5.37$]

$37.$ Find an equation of the plane that passes through the point $(1, -2, 1)$ and contains the line of intersection of the planes $x + y - z = 2$ and $2x - y + 3z = 1$. $\bbox[3px,border:2px solid ...
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3answers
316 views

Sum of closed subspaces of normed linear space

Problem Suppose $R$ is a normed linear space, then show that: If $M$ is closed subspace of $R$ and $N$ a finite dimensional subspace of $R$, then the set $$M+N=\{ z : z = x + y , x \in M , y \in N ...
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vote
1answer
185 views

Let $W \subset V$ vector spaces, let $W'$ be the orthogonal complement of $W$. Show $\dim(W)+\dim(W')=\dim V$

As title says: Let $W \subset V$ vector spaces, let $W'$ be the orthogonal complement of $W$. Show $\dim(W)+\dim(W')=\dim V$. We are given that $W \subset V$ finite vector spaces, symmetric bilinear ...
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2answers
55 views

Direct sum of subspaces of the three dimensional space

$\newcommand{\span}[0]{\mathrm{span}}$I got stuck showing the following problem: If $\mathbb{R}^3 = W\oplus U$ where $W=\span\{e_1\}$ then $U = \span\{e_2,e_3\}$ I tried this way: Since ...
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vote
1answer
46 views

Choosing the right isomorphisms

The question makes sense in every abelian category, but for the moment let's work in the category of vector spaces over a field. Suppose we have two exact sequences $$ 0\to A \to B \to C \to D \to E ...
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vote
1answer
202 views

Calculating a basis of vector space $U \cap V$

So I have two vector spaces: $ U := \langle(1,2,1,2), (1,2,3,3), (1,2,2,3)\rangle $ and $ V := \langle(2,0,2,1), (3,2,3,2), (0,4,0,1)\rangle $ I was able to calculate the base of both $U$ and $V$: $ ...
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vote
1answer
44 views

Calculations in $K$-Algebras

Suppose we have some field $K$ and non-zero elements $a,b,$ in $K$. Define $H=H(a,b)$ to be the $K$-algebra with basis $\{1,x,y,z \}$ over $K$ satisfying $$x^2=a, \\ y^2=b, \\ z=xy=-yx$$ Question: How ...
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3answers
239 views

A Vector Space is a Set - Axiom or Derivation?

I understand that structures with the properties of the real and complex numbers can be defined and derived from the axioms of ZFC set theory. But can a structure with the properties of a (possibly ...
1
vote
1answer
668 views

difference between parallel and orthogonal projection

i would like to understand what is a difference between parallel and orthogonal projection?let us consider following picture we have two non othogonal basis and vector A with ...
1
vote
1answer
818 views

Prove that the field F is a vector space over itself.

How can I prove that a field F is a vector space over itself? Intuitively, it seems obvious because the definition of a field is nearly the same as that of a vector space, just with scalers instead of ...
1
vote
2answers
232 views

Show that the area vectors for a general $n$-sided closed shape sum to zero

It is possible to show that the sum of the area vectors for a general, closed, $n$-sided figure in $\mathbb{R}^3$ (3-space) is zero. Hint: it may be easiest to consider orientable and non-orientable ...