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
26 views

Method for finding intersection between two basis

What is the general way to find a basis for the intersection of two sub spaces? There's the method that use the fact that if we take some vector $v\in V$ and $v\in U$ then every linear combination ...
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0answers
29 views

An exercise question on Hoffman's Linear Algebra

Is the vector (3,-1,0,-1) in the subspace of $R^5$ spanned by the vectors (2,-1,3,2), (-1,1,1,-3), (1,1,9,-5)? I think these vectors all live in $R^4$ instead of $R^5$ so they the answer is no, but ...
3
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1answer
39 views

Why does a differential form represent a vector field?

I'm trying to learn the Divergence/Stoke's theorem and I can't wrap my head around the meaning of a differential form in this context. What does it mean that a differential form represents a vector ...
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1answer
40 views

Losing a dimension when finding intersection between subspaces

Let $F=\mathbb Z_3, V=F^4$. Let $U=sp\{(1,0,0,0),(1,0,1,0),(0,1,1,1) \} \\W=sp\{(0,0,1,0),(-1,1,0,1),(1,1,1,1) \}$ Find $dim (U\cap W)$ we have $v\in U \text{ and } v\in W$ so $v=v$ ...
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1answer
16 views

Short question about modulo space $\mathbb Z^n_p$ and the zero vector

Say we have a vector in $\mathbb Z^3_5$: $v= (1,2,0)$ it looks like it isn't the zero vector but if we multiply it by a scalar: $5v=(5,10,0)\overset{mod5}=(0,0,0)$ so now it is the zero vector and we ...
2
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1answer
45 views

A linear algebra question

$V$ is a vector space with finite dimension. Let $f_1, \ldots,f_m\in\operatorname{End}(V)$ be linear maps of $V$ to itself. Suppose that $V=\ker(f_1)+\ldots+\ker(f_m)$. Show that there are $g_1, ...
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1answer
33 views

A question on vector subspace [duplicate]

Let $V$ be the vector space of all functions $f \colon \mathbb{R} \to \mathbb{R}$ over $\mathbb{R}$, is the set of functions which are continuous a subspace? I think if you add functions which are ...
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1answer
36 views

A excercise problem on Hoffman Linear Algebra

Let $V$ be the vector space over $\mathbb R$ of all functions $f :\mathbb R \to\mathbb R$, then identify if the following is a subspace of $V$: All $f \in V$ such that $f(x^2)=f(x)^2$ While I ...
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2answers
16 views

rotating linear dependent vectors in space

I'm not quite sure how to write this succinctly with mathematical symbols, so I just had to write it out in english. Any edit to suggest how to write it in mathematical form would be appreciated even ...
2
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1answer
45 views

Prove that the product of 2 vectors Normally distributed converges for large dimensions to the full zero matrix

Let $\mathbf{x}, \mathbf{y}$ $\in C^{M \times 1}$ are two i.i.d. vectors with distribution $\mathcal{CN(0,1)}$. How we can prove by the strong law of large numbers that: $\lim_{M\rightarrow \infty} ...
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1answer
18 views

About finding intersection between two vector spaces

Let $W=sp \{e_1,e_2,e_3,e_4\}, U= sp\{(1,-2,1,0),(0,3,-1,1)\}$ be vector spaces both are linearly independent. Show that $U\cap W = sp\{(3,0,1,2)\}$. I know that $\dim U\cap W =1$. Now ...
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1answer
15 views

Subvector and related subspace

This might be easier than I think, but I got stuck. Assume a vector $y=[y_1,\ldots,y_n]\in Y$, where $Y$ is a convex polyhedron. Assume a $k$-dimensional subvector of $y$, namely ...
2
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1answer
57 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 ...
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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
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1answer
61 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 $\Bbb R$. $\mathcal P$ ...
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1answer
33 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
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1answer
128 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 ...
1
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2answers
42 views

Is 'basis times square matrix' a new basis?

Suppose we have a vector space $V = (K, +, \cdot)$. Let $B$ be a basis of 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 ...
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1answer
19 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 ...
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0answers
38 views

Finding a basis for $R^2$ with some constraints

Find a Basis $B$ of $R^2$ s.t. (1) $\left(\begin{array}{c}1 \\2\end{array}\right)_B = \left(\begin{array}{c}3 \\5\end{array}\right)$, and (2) $\left(\begin{array}{c}3 \\4\end{array}\right)_B = ...
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1answer
15 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, $0$ ...
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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
73 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 ...
0
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1answer
18 views

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
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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
17 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
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2answers
103 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 ...
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1answer
33 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
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2answers
75 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 ...
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4answers
74 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
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1answer
26 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
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1answer
19 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, ...
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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
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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 + ...
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0answers
31 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 ...
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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 ...
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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
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1answer
28 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 ...
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1answer
20 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) ...
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1answer
24 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 ...
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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
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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 ...
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1answer
27 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 ...
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1answer
21 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
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1answer
24 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
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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. ...
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2answers
17 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
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2answers
31 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
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1answer
28 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\\ ...
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3answers
198 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\;,\; ...