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 ...

learn more… | top users | synonyms

5
votes
2answers
10k views

plane determined by 2 vectors

i have 2 perpendicular vectors in space . How can i determine the plane determined by the 2 vectors? Regards, Alexandru Badescu
5
votes
3answers
60 views

Vector-space almost linear function

Find a function $f:V \rightarrow W$, where $V$ and $W$ are vector spaces (and V is defined on $\mathbb{K}$), such that $$f(x+y) = f(x) + f(y), \forall x,y \in V$$ but $$\exists a \in \mathbb{K}: ...
5
votes
2answers
16k views

Convert angle (radians) to a heading vector?

I have been looking everywhere trying to find out how to convert an angle in radians (expressed as -Pi to Pi) to a heading vector. The only [x,y] answer I have found is, [cos(angle), sin(angle)] , ...
5
votes
2answers
1k views

Power-reduction formula

According to the Power-reduction formula, one can interchange between $\cos(x)^n$ and $\cos(nx)$ like the following: $$ \cos^n\theta = \frac{2}{2^n} \sum_{k=0}^{\frac{n-1}{2}} \binom{n}{k} ...
5
votes
6answers
694 views

Given ${u, v, w}$ is a basis for $\mathbb{R}^3$, how can I show that $\{u + v + w, v + w, w\}$ is also a basis?

Given ${u, v, w}$ is a basis for $\mathbb{R}^3$, how can I show that $\{u + v + w, v + w, w\}$ is also a basis? I solved a similar problem in $\mathbb{R}^2$ (or at least think I did :p). ...
5
votes
4answers
9k views

Linear Algebra: determine whether the sets span the same subspace

So I am stuck on 51 here: 51. Determine whether the sets $S_1$ and $S_2$ span the same subspace of $\mathbb{R}^3$: $$\begin{align*} S_1 &= \Bigl\{ (1,2,-1),\ (0,1,1),\ (2,5,-1)\Bigr\}\\ ...
5
votes
4answers
691 views

Isomorphism of Vector spaces over $\mathbb{Q}$

From this post we see that $\mathbb{R}$ over $\mathbb{Q}$ is infinite dimensional. Similarly $\mathbb{C}$ over $\mathbb{Q}$ is also infinite dimensional, and I rememeber having solved a problem that ...
5
votes
2answers
51 views

Let $A_{j,k} = \langle x_j, x_k\rangle$. Show $A$ is invertible if and only if $x_1, \ldots, x_n$ are linearly independent.

Let $V$ be a vector space over $\mathbb C$ with inner product $\langle, \rangle$ and let $x_1, \ldots, x_n$ be vectors in $V$. Consider the $n \times n$-matrix $A$ with entries $A_{j,k} = \langle ...
5
votes
1answer
157 views

Let $a_1, …,a_n , b_1,…b_n$ be $2n$ distinct elements of a field , then is the matrix $\Big(\dfrac1{a_i-b_j}\Big)_{ij}$ non-singular?

Let $a_1, ...,a_n , b_1,...b_n$ be $2n$ distinct elements of a field and define $$h_{ij}:=\dfrac1{a_i-b_j} , \forall i,j=1,2 ,\dots,n. $$ Is the $n \times n$ matrix $H:=(h_{ij})$ non-singular ?
5
votes
2answers
66 views

If some vectors in $\mathbb Q^n$ are linearly independent over $\mathbb Q$ , then are they also linearly independent over $\mathbb C$?

Let $\vec v_1 , ..., \vec v_k $ be vectors in $\mathbb Q^n$ linearly independent over $\mathbb Q$ , then is it true that $\sum_{i=1}^ka_i\vec v_i=0, a_i\in \mathbb C, \forall 1\leq i\leq k \implies ...
5
votes
2answers
78 views

Necessary condition for have same rank

Let $P,Q$ real $n\times n$ matrices such that $P^2=P$ , $Q^2=Q$ and $I-P-Q$ is an invertible matrix. Prove that $P$ and $Q$ have the same rank. Some help with this please , happy year and thanks.
5
votes
1answer
173 views

Why is the × operator not defined for vector × vector?

For a vector space, the + operator maps two vectors to another vector while the × operator maps a scalar and a vector to another vector. To me, it seems strange that scalars are seen as separate to ...
5
votes
1answer
400 views

What is the meaning of superscript ⊥ for a vector space

This should be an easy question, if A is a matrix, then the nullspace of A is a vector (sub)space. Then, what is the meaning of superscript inverted T on a vector (sub)space? e.g. $(\mathrm {nullspace ...
5
votes
2answers
85 views

linear map $f:V \rightarrow V$, which is injective but not surjective

I am trying to find a linear map $f:V \rightarrow V$, which is injective but not surjective. I always thought that if the dimension of the domain and codomain are equal and the map is injective it ...
5
votes
1answer
115 views

Must a normed vector space be over $\mathbb{R}$ or $\mathbb{C}$?

If it must be, why is this so? In the maths courses I have taken normed vector spaces always have been over $\mathbb{R}$ or $\mathbb{C}$, but I don't see that this has to be so. I am asking because I ...
5
votes
3answers
861 views

Why orthogonal basis?

Lets take the $\mathbb{R}^3$ space as example. Any point in the $\mathbb{R}^3$ space can be represented by 3 linearly independent vectors that need not be orthogonal to each other. What is that ...
5
votes
2answers
111 views

Linear operators on $C^\infty[a,b]$

I do not know too much about linear operators so forgive me if this doesn't make much sense, but what would the space of linear operators on $C^\infty[a,b]$ be defined as? If we denote this space as ...
5
votes
3answers
257 views

What “is” a matrix in the context of a vector space?

I'm familiar with the definition of a vector space $V$ over a field $F$ I'm also comfortable with the notion that a matrix "represents" a linear map from one vector space $V$ to another vector space ...
5
votes
1answer
233 views

Is there a difference between abstract vector spaces and vector spaces?

I am following my Oxford syllabus and my next step is abstract vector spaces, in my linear algebra book I've found vector spaces. I've searched a little and made a superficial comparison between ...
5
votes
3answers
31k views

Find the equation of the plane passing through a point and a vector orthogonal

I have come across this question that I need a tip for. Find the equation (general form) of the plane passing through the point $P(3,1,6)$ that is orthogonal to the vector $v=(1,7,-2)$. I would ...
5
votes
1answer
260 views

A problem : subspaces of a vector space

Let $L$, $M$, and $N$ are subspaces of a vector space. Prove that following is not necessarily true. $L \cap (M + N) = (L \cap M) + (L \cap N) $ This problem is given in 'Finite dimensional vector ...
5
votes
2answers
319 views

Riemann Sphere and a Strange Vector Space Definition

I'm reading Fractals Everywhere by Michael Barnsley. On pp. 6-8 [1] he defines a linear space which, he says, "is also called a vector space." However, his definition of a linear space only requires ...
5
votes
2answers
185 views

Can we have something like an orthonormal basis for a finite dimensional normed space?

So I proved a certain theorem about finite dimensional inner-product spaces, but after completing the proof, I realized the only point where I used the idea of orthogonality was the construction of an ...
5
votes
2answers
145 views

Are there norms on $\Bbb{C}^m$ and $\Bbb{C}^n$ so that the norm $\Vert\cdot\Vert$ is a subordinate norm?

Denote $$\Vert A\Vert=\sum_{1\le j,k\le m}\vert A_{j,k}\vert$$ is cleary a norm over $M_{m,n}(\Bbb{C})$ but not a subordinate norm by taking the identity matrix $I$. So my question is: Can we make ...
5
votes
1answer
63 views

What series of 'hyperpolyhedrons' do exist? Is there an effective way to derive their cross-sections by 3-d subspace?

There are two obvious series of 'hyperpolyhedrons'. 'Hyperoctahedron' with vertices $(\pm1,0...0), (0,\pm1,0,...0)...(0,...0,\pm1)$ and each vertex connected by an edge with each other vertex ...
5
votes
1answer
158 views

Intuition for “the existence of a basis for every vector space is equivalent to the Axiom of Choice”?

Is there a intuitive way to understand "the existence of a basis for every vector space is equivalent to the Axiom of Choice"?
5
votes
2answers
77 views

Projection into a subspace?

Let $S$ be a nonzero subspace with orthogonal basis $(v_1, \ldots, v_k)$. Then the projection of $u$ onto $S$ is given by: $$\operatorname{proj}_S u = \frac {v_1 \dot{} u}{\operatorname{norm} ...
5
votes
3answers
174 views

Find the necessary and sufficient conditions on $A$ such that $\|T(\vec{x})\|=|\det A|\cdot\|\vec{x}\|$ for all $\vec{x}$.

Consider the mapping $T:\mathbb{R}^n\mapsto\mathbb{R}^n$ defined by $T(\vec{x})=A\vec{x}$ where $A$ is a $n\times n$ matrix. Find the necessary and sufficient conditions on $A$ such that ...
5
votes
1answer
363 views

Adjoint of forgetful functor between category of vector spaces and category of abelian groups

I've just found out about the forgetful functor between the category of vector spaces and the category of abelian groups. It maps a vector space to it's additive abelian group. My question is, is ...
5
votes
3answers
152 views

Finding the inverse of a matrix using a series

I want to find the inverse of the matrix $A$ given by: $ \left( \begin{array}{cc} 1 & -\epsilon \\ \epsilon & 1 \\ \end{array} \right) $ where $|\epsilon|$ $< 1$ (although I do not ...
5
votes
2answers
205 views

Cross products?

Say you have vectors $v$ and $w$. Let there cross product be denoted by $x$ so that: $$v \times w = x$$ According to Wikipedia: $$x_x = v_yw_z - v_zw_y$$ $$x_y = v_zw_x - v_xw_z$$ $$x_z = v_xw_y - ...
5
votes
2answers
2k views

Proof of equivalence of algebraic and geometric dot product? [duplicate]

Geometrically the dot product of two vectors gives the angle between them (or the cosine of the angle to be precise). Algebraically, the dot product is a sum of products of the vector components ...
5
votes
3answers
2k views

Is the zero vector in $\mathbb{R}^n$ by itself a subspace of $\mathbb{R}^n$?

W is a subspace of $\mathbb{R}^n$ iff The zero vector ∈ W. X + Y ∈ W for any X, Y ∈ W. aX ∈ W for any X ∈ W and a ∈ R. So, given W = { X : X = [x1...], x1 = 0, x2 = 0, ... xn = 0 } ∈ Rn The zero ...
5
votes
2answers
247 views

How to get the upper triangular form for any linear map?

Let $T:V \to V$ be linear. $V$ is a complex vector space of dimension $k$. Then there exists a basis so that the matrix generated by $T$ under that basis is upper triangular. The proof is by induction ...
5
votes
2answers
417 views

Probability that $n$ vectors drawn randomly from $\mathbb{R}^n$ are linearly independent

Let's take $n$ vectors in $\mathbb{R}^n$ at random. What is the probability that these vectors are linearly independent? (i.e. they form a basis of $\mathbb{R}^n$) (of course the problem is ...
5
votes
2answers
1k views

What matrices preserve the $L_1$ norm for positive, unit norm vectors?

It's easy to show that orthogonal/unitary matrices preserve the $L_2$ norm of a vector, but if I want a transformation that preserves the $L_1$ norm, what can I deduce about the matrices that do this? ...
5
votes
4answers
128 views

Vector dimension of a set of functions

Let $F$ be a field and $S$ an infinite set. Set $V=\{f:S \rightarrow F\}$ endowed with the vector space structure that results from the pointwise operations of $F$. It is easy to prove that $|S| \leq ...
5
votes
1answer
823 views

direct products and direct sums for matrices and for vector spaces

I was wondering what relations and similarities are between direct product for matrices and direct product for vector spaces? Or do they just unfortunately and somehow misleadingly happen to have the ...
5
votes
1answer
96 views

What is needed to make Euclidean spaces isomorphic as groups?

Consider the abelian groups $G_n=(\mathbb R^n,+)$ for $n\geq1$. Claim: For any $n$ and $m$ the groups $G_n$ and $G_m$ are isomorphic. This claim is true if one assumes the axiom of choice, and I ...
5
votes
1answer
145 views

Is it possible to define an inner product such that an arbitrary operator is self adjoint?

Given a vector space $V$ (possibly infinite dimensional) with inner product $(.,.)$. We say an operator $A$ is self adjoint if $(Af,g)=(f,Ag)$. The definition as stated require us to start with an ...
5
votes
1answer
52 views

Is this condition sufficient to determine the linear space is of finite dimension?

From the Banach theory we knew that: 1) A linear space(a vector space endowed with its vector topology) $X$ of finite dimesion $dimX=n$ has the following property: If ${\left\| \bullet \right\|_1}$ ...
5
votes
2answers
85 views

Is there a name for the set of bit combinations of bitstrings?

Let $A \subset \{0,1\}^n$ be a set of $n$-bit bit vectors. Let me call a bit vector $b = (b^{(1)}, b^{(2)}, \dotsc, b^{(n)}) \in \{0,1\}^n$ a "bit combination" of the vectors in $A$ if: $$\forall i ...
5
votes
1answer
72 views

Generator of End(V)

If V is a finite-dimensional vector space of dimension n and G⊂End(V) such that G generates End(V) meaning that any element of End(V) is expressible as a linear combination of products of a number of ...
5
votes
1answer
4k views

How do you determine whether a given set of functions is a subspace of C[-1,1]?

I'm having a terrible time understanding subspaces (and, well, linear algebra in general). I'm presented with the problem: Determine whether the following are subspaces of C[-1,1]: a) The set ...
5
votes
3answers
143 views

Do all vectors belong to a vector space?

If we were given the vector $(1,1,1)$, say, we know immediately that it belongs to the vector space $\mathbb{R}^3$ (and infinitely many others). But, if we take this vector of dolphins: or some ...
5
votes
1answer
120 views

Are coordinate projections continuous?

Okay I have been working under the assumption that this is "obvious" for a while now, but it started to bug me and now I'm fumbling to prove it. Suppose $X$ is a normed linear space (possibly ...
5
votes
3answers
2k views

Question on finite Vector Spaces, injective, surjective and if $V$ is not finite

Let $V$ be a vector space and $\alpha \in \operatorname{End}(V)$ (i) If $V$ is finite dimensional, then $\alpha$ is injective iff $\alpha$ is surjective. (ii) Give example showing (i) is false if ...
5
votes
1answer
58 views

When is it true that $\dim(U \cap (V+W))=\dim(U \cap V + U \cap W)$?

I apologize if this is a silly question( which may have been asked before), I was wondering after seeing a post on this list on math-overflow When is it true that $\dim(U \cap (V+W))=\dim(U \cap V + ...
5
votes
3answers
69 views

A direct proof on if $(X, ||\cdot ||)$ is a normed vector space and $Y\subset X$, with $Y$ having finite dimension, then $Y$ is closed.

I am trying to produce a direct proof on the statement mentioned above. The field I am working in is $\mathbb{R}$. My proof outline goes as following: If $Y$ is finite-dimensional, there exists a ...
5
votes
1answer
114 views

Duality mappings on finite-dimensional spaces

I have a few questions regarding some concepts in a book "nonlinear partial differential equations by Roubicek" I am studying. The following is from the text. "Let $V$ be a separable, reflexive ...