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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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 ...
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1answer
172 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 ...
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1answer
657 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 ...
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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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2answers
40 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 ...
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2answers
438 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.
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1answer
105 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 $\mathbb{R}^3$, $E=\{e_1,e_2,e_3\}$ ...
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4answers
330 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
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4answers
181 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
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1answer
417 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
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2answers
92 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
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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
172 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
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0answers
213 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 ...
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1answer
51 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
80 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 ...
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1answer
18 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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0answers
52 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 ...
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1answer
41 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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2answers
102 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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3answers
68 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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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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4answers
107 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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2answers
50 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
112 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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1answer
308 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
298 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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1answer
179 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
54 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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1answer
45 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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1answer
190 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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2answers
64 views

Unitary map between sets of vectors

Suppose I have two sets of vectors, $E_1=\{v_i\}_{i=1}^{k}$ and $E_2=\{u_i\}_{i=1}^{k}$, with each vector belonging to $\mathbb{C}^k$. When is it possible to find a unitary matrix that maps $E_1$ to ...
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1answer
43 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
233 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 ...
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1answer
596 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 ...
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1answer
768 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 ...
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2answers
217 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 ...
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2answers
129 views

Inconsistent System of Linear Equations

Let $A ∈ M_{n\times n}(F)$. Suppose that the system of linear equations $AX = B$ has more than one solution. Prove that there is a column $C ∈ F^n$ such that the system of linear equations $AX = C$ is ...
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1answer
112 views

angle between steepest gradient of two plane

IF I have two 3d planes such as Oab and Oa'b'. If these two planes intersect a horizontal plane and the intersection of each plane makes AB and A'B' lines. then, Does the angle between AB, A'B' ...
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0answers
188 views

Vector Projection and Cross Product

Let there be $w, u, $ and $v$, such that: $$w \times u = \langle1, 3, 5\rangle$$ $$w \times v = \langle 2, 4, 6\rangle$$ Find: $$v \cdot (((u \times w) \times v) + \text{VP}uv(w)) + ((u ...
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1answer
84 views

Well-definedness of a linear mapping

I have the following theorem: Theorem (from Schaum's Linear Algebra) Let $V$ and $U$ be vector spaces and $\{v_1, \ldots, v_n\}$ be a basis on $V$. Let $\{u_1,\ldots, u_n\}$ be arbitrary vectors ...
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1answer
170 views

Orthonormal basis with three vectors

So I have the following question: $Let \\ u_1 = \begin{pmatrix} \frac{1}{3\sqrt{2}}\\\frac{1}{3\sqrt{2}}\\\frac{-4}{3\sqrt{2}}\end{pmatrix} u_2 = \begin{pmatrix} ...
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vote
4answers
437 views

Get the direction vector passing through the intersection point of two straight lines

Let say I have this diagram, How to find the direction vector passing through the intersection point of two straight lines? Update: new vector is the bisector of two lines and vector may be ...
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1answer
190 views

Finite dimensional vector space with subspaces [duplicate]

Possible Duplicate: Could intersection of a subspace with its complement be non empty. Is it possible for a finite dimensional vector space to have 2 disjoint subspaces of the same ...
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1answer
170 views

What is an honest basis?

In a comment to this question, the commentator stated that "the monomials form an honest basis for your vector space". To be honest, I never heard of that. Is this something elementary?
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1answer
154 views

Is this argument valid? — bilinear maps

Does this argument make sense? So I have a bilinear form $B:V\times V\to F$. (BTW, is bilinear form equivalent to a bilinear map?) where $V$ is a finite-dimensional vector space. I have a subspace ...
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1answer
744 views

Show that this is a vector space and determine the dimension

Question: Let $v = (1,1,0,1) \in \mathbb{R}^{4}$ Let $$V:= \{f: \mathbb{R}^{4} \rightarrow \mathbb{R}^{2} \mbox{ linear } | f(v) = 0\}.$$ (a) Show that $V$ forms a vectorspace. (b) Determine ...
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1answer
30 views

Yet another n-Dimensional Rotation question: is there a definition for n-Dimensional rotation which is cosine-distance invariant? (TO CLOSE)

(Moved to Searching for a definition for n-Dimensional rotation which is cosine-distance invariant, flagged it to delete it) I'm wondering if there exists a rotation definition by which the vectors ...
0
votes
1answer
23 views

Scalar Products equation proof

$\langle \langle x + y, z \rangle \rangle = \langle \langle x, z \rangle \rangle + \langle \langle y, z \rangle \rangle$ It is clear when there are only $\langle \dot \ , \dot \ \rangle$ but what ...