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

2
votes
1answer
62 views

Vector Calculus Operator $\vec{u} \cdot \nabla$

I just want to double check on this operator and it's properties. It pops up in fluid mechanics often and I just want to be sure about my understanding: 1) $$(\vec u \cdot \nabla)\vec u$$ Is this ...
1
vote
2answers
44 views

How to check is two subspaces are the same.

Suppose I have some $N$ dimensional real vector space and two $M<N$ dimensional subspaces of that, and say I know one set a basis vectors for each: ${v_i}$ where $i=1,2,...,M$ and ${w_i}$ where ...
2
votes
1answer
64 views

$\operatorname{span}(x^0, x^1, x^2,\cdots)$ and the vector space of all real valued continuous functions on $\Bbb R$

Let $p_n(x)=x^n$ for $x\in\Bbb R$ and let $\mathcal P=span\{p_0,p_1,p_2,\cdots\}$ . Then $\mathcal P$ is the vector space of all real valued continuous functions on $\Bbb R$. $\mathcal P$ ...
0
votes
1answer
36 views

Hilbert space isometric to a subspace of its dual

Let $\cal H$ be a Hilbert space, and let $\cal H^\ast$ be its dual (of the continuous functionals). If $\cal H$ is a real vector space, I can define: $$\begin{align}\Phi\colon\, &{\cal H} \to ...
6
votes
1answer
130 views

Prove $AB=I$ implies $BA=I$ using Fitting's Lemma

I know this question has already been asked, but I need a proof that for $A,B \in M_n(K)$, $$AB=I_n \Rightarrow BA=I_n$$ using Fitting's lemma. I thought of using the fact that $K^n$ is a ...
0
votes
2answers
57 views

Is “basis times square matrix” a new basis?

Suppose we have a vector space $V = (K, +, \cdot)$. Let $B$ be a basis for $V$. Now we take an arbitrary square matrix $S \neq 0$. $BS$ is just a linear combination of $B$. Thus $BS$ should be a new ...
1
vote
1answer
21 views

Vector space generated by set intersection

Again, I've come across a simple task, but being new to linear algebra, I wish not yet to question my textbook's author credibility. In vector space $R^4$ , two subspaces $W_1$ and $W_2$ are generated ...
1
vote
1answer
17 views

Linear mapping from square matrix vector space to polynomial vector space

Let $M_2(\mathbb{R})$ be a vector space of all 2 dimensional square matrices and $P$ be the space of all $2^{nd}$ degree polynomials. Suppose we have a linear mapping defined as ( from here on, $0$ ...
1
vote
4answers
55 views

Prove that $\{x^2+x , x^2-1, x+1\}$ generates vector space of $2^{nd}$ degree polynomials

I am quite new to linear algebra and am having some trouble with the abstractness of some it's parts. For example, this task seems quite simple and as if there's no need to do any proving, but to ...
4
votes
2answers
79 views

Proof that $\mathbb{R}^+$ is a vector space

I was doing some beginner linear algebra tasks and stumbled upon this one: Proove that $\mathbb{R}^+$ is a vector space over field $\mathbb{R}$ with binary operations defined as $a+b = ab$ (where ...
1
vote
1answer
22 views

Possible inconsistency of column representation with orthogonality of vectors

Let's say I have two vectors $v_{1}$ and $v_{2}$ which form a basis for $\mathbb{R}^2$. Any vector $v$ in $\mathbb{R}^2$ can be represented as $$v = av_{1} + bv_{2}$$ for some $a,b \in \mathbb{R}^2$. ...
1
vote
1answer
24 views

Proving parallel planes in $\mathbb{R}^4$

Given two planes in $\mathbb{R}^4$ (or perhaps higher dimensions) in parametric form, what ways are there to prove that they are parallel (or not parallel)? A friend suggested equating the spans and ...
0
votes
2answers
18 views

Is equation of a hyperplane fixed?

If I have a $n$ dimensional vector space ( real components ) then a hyperplane will be $n-1$ dimensional. The equation of a hyperplane is defined as $\vec{n}.\vec{x}=\vec{n}.\vec{x_0}$ ( if I am not ...
4
votes
2answers
106 views

Prove that $\dim(V)$ is even

Let $V$ be a finite dimensional vector space. Let $A_1,A_2: V\rightarrow V$ be commuting linear operators such that $A_1+A_2=-I$ where $I$ is the identity operator. Also $A_1,A_2$ have no negative ...
5
votes
1answer
37 views

Subspaces of $\Bbb R^n$ containing vectors whose coordinates satisfy prescribed inequalities

For any integer $n\ge2$, the vector space $\Bbb R^n$ is divided into $n!$ "wedges" by prescribing which coordinate is largest, second-largest, etc. One such wedge is $$\{(x_1,\dots,x_n)\in\Bbb ...
2
votes
2answers
79 views

Regular Quadratic Space - isotrope vector

I am currently trying to solve the following exercise: Show that every regular quadratic space of finite dimension $E$ that contains at least one isotrope vector, has a basis consisting only of ...
1
vote
4answers
77 views

Why are there infinitely many orthonormal vectors?

By Graham Schmidt process we can create infinitely many orthonormal vectors, but my doubt is that why is it not bounded by the dimensionality of the space ? Intuitively (geometrically) how can we ...
1
vote
1answer
28 views

If $\dim(V_F)$ is Infinite, Does It Follow $\dim(\operatorname{Hom}(V_F, W_F)) \ge |F|$?

Part of the proof that $\dim(V^*_F) > \dim(V_F)$ for an infinite dimensional space is that $\dim(\operatorname{Hom}(V_F, F)) \ge |F|$ (i.e $\dim(V^*_F) \ge |F|$). See for example Dual space ...
0
votes
1answer
20 views

How to determine whether a vector is in the span of a a set of vectors modulo 2?

I already found out how to check whether a vector is contained in a set of vectors using Gaussian Elimination / RREF. My problem is that I can't find a way, even after researching for several hours, ...
0
votes
2answers
31 views

Find Orthogonal Vector's Peak Point

I am given a 3-component vector $\vec v$. There are obviously an infinite number of orthogonal vectors to $\vec v$. I need to find the specific orthogonal vector, lets call it $\vec{x}$, in the plane ...
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 + ...
1
vote
2answers
72 views

Why people use the Gram-Schmidt process instead of just chosing the standard basis

I really can't find a reason for going through all the work of the Gram-Schmidt method to make a new orthogonal basis $B'$ given an old basis $B$. If I want to change to an orthogonal basis, the most ...
1
vote
0answers
33 views

For Vector Spaces V and W with one Infinite Dimensional , is Hom(V, W) Isomorphic to Hom(W, V)?

If V and W are both finite then clearly Dim (Hom(V, W)) = Dim(V).Dim(W) = Dim(Hom(W, V)) so they are isomorphic. I'm not so sure if one is infinite. An "infinite matrix" construction for a linear ...
1
vote
1answer
21 views

Proof that sum of subspaces gives $\mathbb{R}^{n}$

Let A be square matrix of order $n\geq2$ such that $A^{2}=I$. Prove that $\mathbb{R}^{n}=U\oplus W$, where $U=\{x\in\mathbb{R}^{n}:\; Ax=x\},\; W=\{x\in\mathbb{R}^{n}:\; Ax=-x\}$. To prove, I ...
1
vote
1answer
20 views

Matrix Multiplication: Connection between contexts.

I've been thinking about two well understood uses for matrix multiplication: 1) Composition of Linear Maps. Let $T,U$ be endomorphisms of a vector space $V$, and let $A,B$ be their respective ...
1
vote
1answer
33 views

dimension formula problem

Let $V$ be a finite-dimensional vector space and let $A_1$, $A_2$, $B_1$, $B_2$ be subspaces of $V$ such that: $$\dim A_1 = \dim A_2$$ $$\dim B_1 = \dim B_2$$ and $$A_1 + B_1 = A_2 + B_2 = V$$ Show ...
0
votes
1answer
21 views

Linear Algebra- Sums of Vector Spaces

I dont know how to prove this although intuitively I know that it is true: Let $ V $ be a finite dimensional vector space and $S$ and $T$ be subsets of $ V $. Show that $$ Sp(S\cup T) = Sp(S)+Sp(T) ...
0
votes
1answer
26 views

A unitary space, interpreted as a Euclidean space

Let $(V, \gamma)$ be a $n$-dimensional unitary space. Let $V_{\mathbb{R}}$ be the vector space $V$, interpreted as a $2n$-dimensional $\mathbb{R}$-vector space. I first want to show that ...
0
votes
2answers
22 views

Vector subspaces of functions

Let $F(R)$ be the set of all functions. Which the following subsets of $F(R)$ are vector subspaces? (i) $S_1 = \{f \in F(R) : f(\sqrt{2})=0\}$ (ii) $S_2 = \{f \in F(R):f(x)=0, x \in R\}$ (iii) $S_3 ...
0
votes
1answer
32 views

Show a set of vectors is a basis for the space of linear transforms

Define: f: R^n->R by: f(x)=ei*x, where ei is the i-th basis vector in R^n {f1,f2,..,fn} Suppose: c1f1+c2f2+...+cnfn=0 Now I know that if I feed the fs ei, i will be left with Ci=0. This is ...
0
votes
1answer
30 views

Vector spaces, bases and components

Consider the usual vector space $\mathbb{R^2}$. The vectors are ordered couple of real numbers. The ordered couple as a whole is a vector in $\mathbb{R^2}$, the first and second elements of the ...
1
vote
1answer
24 views

Inner product axioms

Is there a shortcut to finding out if a particular operation is an inner product? Applying the axioms takes a long time especially when in exams so is there a quick way to find out if the operation is ...
0
votes
1answer
25 views

Projecting a vector on orthogonal planes

I am looking from an engineer point of view. I have a sensor for which I need vector projecting on two different planes. I have the unit vector in the body frame that is to be projected and I obtained ...
2
votes
1answer
20 views

Do central isometries on complex spaces respect addition?

It can be proved that Any central isometry on $\mathbb{R}^n$ is a linear transformation. So I was wondering whether central isometries on $\mathbb{C}^n$ are also linear transformations. ...
1
vote
2answers
18 views

How many different parallelograms can be drawn if given three co-ordinates in 3D Cartesian vector?

By different, I mean the angles that each parallelograms make are different, the magnitude of the vectors that make each one are different, etc... I had this question on a test, where we have to ...
0
votes
2answers
32 views

Can you have an operator on a vector space such that it is injective but its kernel is not the zero element?

Take any vector space $V$ and an operator $T : V \mapsto V$ Can there exist a $T$ such that it is injective but $\ker T \neq \{0\}$ and equal to some other element instead?
2
votes
1answer
29 views

Is it possible to define a linear transformation piecewise as different functions?

I'm trying to proof that it is not possible. I suspect that if a linear transformation $T:\mathbb{U}\rightarrow\mathbb{V}$ can be defined such that: $$ T(v)=\left\{\begin{matrix} F(v) & v\in A\\ ...
1
vote
3answers
204 views

Find the nullity of a linear operator

Suppose that $V$ is a vector space and $x_1,x_2,\dots,x_n$ is a basis for $V$ and $T:V\rightarrow V$ is a linear transformation such that $$T(x_1)=x_2\;,\; ...
0
votes
1answer
19 views

Calculating new angle after bounce

I'm having a simple question about bouncing, but I can't seem to solve it. I'm having an object that approaches a vector and collides. Now I would like to calculate the new angle after the bounce. I ...
0
votes
0answers
11 views

Operators with infinite number of invariant subspaces

When dimension of a space is more than 1, then identity operator and zero operator has infinite number of invariant subspaces. I wonder, if there are other operators that have infinite number of ...
0
votes
0answers
31 views

Completion of C(I) to $L^{2}(I)$ for some arbitrary interval I

As $L^{2}(I)$ is the completion of C(I), without too many issues (as $L^{2}(I)$ is a space of equivalence classes with equivalence relation defined as functions equivalent if differ at only finitely ...
0
votes
0answers
29 views

Basis for the linear spacer $\ell^p$ [duplicate]

Is there any well known basis (Hamel basis) for the vector space $\ell^p$? And what about the cardinality of such basis? Is it countable?
1
vote
1answer
18 views

Finding a matrix representation of the linear transformation $T\colon P_2\to P_2$ ($T(f) = f''+2f'-f$)

Find a matrix representation of the linear transformation $T: P_2( \mathbb{R} ) \to P_2(\mathbb{R} )$, where $T$ is defined as $T(f(x)) = f''(x)+2f'(x) -f(x)$. I know the standard ordered basis ...
2
votes
2answers
38 views

Every subspace of a vector space has a complement

I want to see if my proof is true or I thought very trivially? If $H$ is a subspace of a finite dimensional vector space $V$, show there is a subspace $K$ such that $H\cap K=0$ and $H+K=V$ So ...
1
vote
1answer
22 views

How to find the closest line to two segments?

I have two segments in 2D space, defined by their endpoint x and y coordinates. How can I find a best-fit line using vector algebra (formally, that minimizes the integral of square-distance from it to ...
3
votes
3answers
71 views

Is this dual pairing the same as the inner product?

If $(V, \langle \cdot$ , $ \cdot\rangle)$ is an inner product space with dual $V^*$ then there is a natural dual pairing $\langle \cdot$ , $ \cdot \rangle ^*: V^* \times V \rightarrow \mathbb K$ given ...
1
vote
0answers
20 views

Calculate pairwise cosine distance only returning the lower triangular matrix

I have a matrix, where each row is a feature vector. I would like to find out the pairwise cosine distance between all of these feature vectors. The cosine value between all rows in a matrix could be ...
4
votes
2answers
108 views

Geometry: What is being calculated here?

Context: I am a computer graphics programmer looking at a code-implementation. I need help understanding a function that has neither been documented properly nor commented. Given a circle with ...
2
votes
0answers
26 views

Get line function after mapping one x|y 2D space to another x|y 2D space

I am not a mathematician, so sorry for my poor dictionary. I have a mapping function of a form: $$f(x,y,w) = \begin{cases}x':\sqrt{x^{2} + y^{2}}\\y':\sqrt{w^{2} - 2wx + x^{2} + y^{2}}\end{cases}$$ ...
-1
votes
1answer
32 views

intersection of two subspaces is not $\{0\}$ [duplicate]

If V and W are 3-dimensional subspace of $R^5$ then prove that V and W must have a nonzero common vector.