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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218 views

Dimension of a subspace of polynomials of a given degree

Let $k[x]=k[x_1,\dots,x_n]$ be the set of polynomials in $n$ variables. Given a $k<n$, how can you determine the dimension of the vector subspace of polynomials of degree $\le k$? I guessed it ...
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2answers
2k views

Point on the left or right side of a plane in 3D space

I have an alpha plane determined by 3 points in space. How can I check if another point in space is on the left side of the plane or on the right side of it? I need a fast solution for plug-in ...
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1answer
48 views

Describe the affine space A made of 5 points and the convex envelope (hull) C

I have 5 points and I want the affine space of those. I also want to describe the convex envelope of the set of points. I searched on google and found how to do the affine space with 3 points. However ...
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1answer
136 views

What is the name of this equation?

I have found this picture but I don't know the name of the equation in it. Another thing, what kind of plots are those in the picture? I have also tried to re copy it: $$ ...
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3answers
138 views

Inner Product Calculation

So the question is: Let $\theta$ be a fixed real number and let $x_1 = \begin{pmatrix} \cos\theta\\ \sin\theta \end{pmatrix}$ and $x_2 = \begin{pmatrix} -\sin\theta\\ \cos\theta \end{pmatrix}$ (a) ...
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1answer
206 views

Vector Project onto Subspace

So the question is: Let S be the subspace of $\mathbb{R}^3$ spanned by the vectors $ u_2 = \begin{pmatrix} \frac{2}{3}\\\frac{2}{3}\\\frac{1}{3}\end{pmatrix} u_3 = \begin{pmatrix} ...
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1answer
172 views

Linear Combination of vectors

I have a previous post here. There is a part b to that question and it asks: Let $x=(1,1,1)^T$. Write x as a linear combination of $u_1, u_2, u_3$ using Parseval's formula to compute $||x||$. I know ...
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1answer
100 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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2answers
690 views

Sets forming orthonormal basis

So the question is: Which of the following sets of vectors form an orthonormal basis for $\mathbb{R}^2$ $(a) \{(1,0)^T, (0,1)^T\}$ $(b) \{(\frac{3}{4},\frac{4}{5})^T,(\frac{5}{13},\frac{12}{13})^T\}$ ...
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3answers
3k views

Union of two subspaces versus intersection of two subspaces

According to the definition, the union of two subspaces is not a subspace. That is easily proved to be true. For instance, Let $U$ contain the general vector $(x,0)$, and $W$ contain the general ...
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0answers
50 views

Knowing $\alpha$ and $\beta$, compute $\gamma$. $vers(\vec v)=(\cos\alpha,\cos\beta,\cos\gamma)$

Knowing that: $\vec v=|\vec v| vers(\vec v)$ $\cos\alpha=\frac{\vec v \cdot \vec i}{|\vec v|\cdot|\vec i|}$ $\cos\beta=\frac{\vec v \cdot \vec j}{|\vec v|\cdot|\vec j|}$ $\cos\gamma=\frac{\vec v ...
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1answer
295 views

Count the number of bases in a subset

Consider $\mathbb{R}^n$ as a vector space over $\mathbb{R}$. Consider the subset $\mathrm{S}^n = \{(x_1,\ldots,x_n) \in \mathbb{R}^n | x_i = 0 \; \mathrm{or} \; 1\;\forall i = 1,\ldots,n\}$. How many ...
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1answer
194 views

How to prove this property for a nilpotent map on a vector space

Let $X$ vector space over $\mathbb{R}$ with $\dim(X)=3$. and let $T\colon X\to X$ be a nilpotent linear map. How can I show that $X$ must have infinitely many $T$-invariant subspaces if and only if ...
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1answer
86 views

What is the defined direction of the curvature vector at a point?

I'm asking this only because I could not find it anywhere on the web (tried Wikipedia and Google searches). The only hint I found was in this image on Wikipedia, which seems to indicate that the ...
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2answers
246 views

Find $\mathbb W$, a subspace, such that…

Consider the following subspaces of $\mathbb R^4$. $$\eqalign{ & \Bbb S = \left\{ {x\in\mathbb R^4:{x_2} - {x_3} - {x_4} = 0} \right\} \cr & \Bbb T = \left\{ {x\in\mathbb R^4:{x_1} + ...
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1answer
249 views

Dot product of outer product of vectors

I'm reading a research paper (for the curious, it's from the field of image processing: Curvature-Based Stroke Rendering by S. Saito et al), and I'm stuck on page 4 trying to understand this equation: ...
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1answer
94 views

Linear Transformation over Subfield

Letting $F\subseteq K$ be fields, and $V$ a vector space over $K$. Certainly, $V$ is also a vector space over $F$. And if $\{e_1,...,e_n\}$ is a basis for $K$ over $F$ and ...
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2answers
63 views

Algebra of Vectors

Is it possible that there are $3$ vectors, $a, b, c$, such that $a + b + c = 0$ but $|a| = 1$, $|b| = 2$ and $|c| = 4$? If yes why? and if no why?. I'm trying to get the solution since last $2$ ...
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0answers
164 views

Divergence Theorem to prove equality of integrals

I'm trying to wrap my head around this problem - the interplay between $\nabla$ and $\Delta$ is doing my head in. It says to use the divergence theorem. Prove that $$\int_\Omega u \cdot \Delta v\, ...
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1answer
476 views

Vector Spaces and AC

I know that the proof that every vector space has a basis uses the Axiom of Choice, or Zorn's Lemma. If we consider an axiom system without the Axiom of Choice, are there vector spaces that provably ...
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2answers
333 views

Can the real field be made a vector space over the complex field?

We all know that $ \mathbb{C} $ is naturally a vector space over $ \mathbb{R} $. However, is there some kind of (possibly weird) scalar multiplication law that would make $ \mathbb{R} $ a vector space ...
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3answers
443 views

Isomorphism Between Vector Spaces

Let $E$ be a infinite dimensional vector space. Let $F\subset E$ be a infinite dimensional subspace. Is it possible that $F$ is isomorphic to $E$? I think this is not possible, but I can't see a ...
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2answers
103 views

existence of hyperplane

Let $V$ be an $n$ dimensional vector space, let $R$ be a finite set of vectors. Will there exist a hyperplane which does not contain any of the vectros from $R$? How to construct such a hyperplane? ...
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1answer
364 views

Splitting Exact Sequences

Given an exact sequence of vector spaces: $$0\longrightarrow U \longrightarrow V \longrightarrow W\longrightarrow 0$$ with $f:U\rightarrow V$ and $g: V \rightarrow W$ I want to prove the that ...
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1answer
322 views

Splicing together Short Exact Sequences

Given an exact sequence of vector spaces $$\cdots\longrightarrow V_{i-1}\longrightarrow V_{i}\longrightarrow V_{i+1}\longrightarrow\cdots$$ I want to show that it is the same as having a collection of ...
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3answers
164 views

Why the morphisms of vector spaces, over different fields is not interesting?

Suppose $V_\mathbb{F_V}$ and $W_\mathbb{F_W}$ are two vector spaces over fields $\mathbb{F}_V$ and $\mathbb{F}_W$. Then a homomorphism of these vector spaces consists of maps $f:V\rightarrow W$ and ...
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2answers
438 views

3D Cartesian Coordinates System revolve around a specified axis

I have a 3d cartesian coordinates system and now I want to rotate a point $p(x_0, y_0, z_0)$ arround a specified axis $v(v_x, v_y, v_z)$ like $(1,1,1)$,and the angle is $\theta$,finally I want to get ...
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1answer
210 views

Using a weighted euclidean inner product to compute $d(u,v)$.

If they gave us $u=(1,1)$, $v=(3,2)$, $w=(-1,0)$ and $k=5$. How do you compute $d(u,v)$? I know that $d(u,v)=\|u-v\|$ but I am lost as how to continue. Please could I get some help on this!
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1answer
468 views

How to plot N points on the surface of a D-dimensional sphere roughly equidistant apart?

Let's say I have a D-dimensional sphere with a radius R. I want to plot N number of points evenly distributed (equidistant apart from each other) on the surface of the sphere. It doesn't matter where ...
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1answer
3k views

How do you find parallel and perpendicular vectors

Given the vectors v and w, write v = v∥ + vp, where v∥ is parallel to w, and vp is perpendicular to w: a) v = (2, 3, -7); w = (1,-2,-5) b) v = (-3, 1, 2); w = (8,5,-3) I found this in an old ...
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1answer
324 views

Prove whether or not the set of all pairs of real numbers of the form $(0,y)$ with standard operations on $\mathbb R^2$ is a vector space?

I just want to make sure I understand this correctly. The problem: Prove whether or not the set of all pairs of real numbers of the form $(0,y)$ with standard operations on $\mathbb R^2$ is a ...
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1answer
673 views

Prove that the set of all quadratic functions whose graphs pass through the origin with the standard operations is a vector space.

The idea is to prove that this is a vector space based upon the following axioms: $\mathbf{u}+\mathbf{v}$ is in $V$. Closure under addition. $\mathbf{u}+\mathbf{v}=\mathbf{v}+\mathbf{u}$. ...
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2answers
1k views

Show that the set of continuous functions $C[a,b]$ is a vector space

I have a question about proving that something is a vector space. For example, I am trying to prove Axiom 6: $$ (A + B)(f(x)) = A(f(x)) + B(f(x)), $$ where $A$ and $B$ are constants. Now, to me, ...
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1answer
404 views

Describe the additive inverse of C($-\infty, \infty$)

I'm just stuck on this question. How can I represent the additive inverse of all continuous functions? The additive inverse: For every $\overrightarrow{u}$ in V, there is a vector V denoted by ...
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1answer
1k views

Cartesian vector to Cylindrical components

I recently started EE 1, and was confused by the jump from cylindrical coordinates to cylindrical vectors In the past, I've just converted between cartesian and cylindrical via: p = sqrt(X^2 + Y^2) ...
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1answer
84 views

Function Vector Spaces: Set to Field

I'm having trouble answering the last problem Linear Algebra set. Not looking for a solution, of course, but some pointers would be incredibly helpful. Given a vector space $F^S$ of all functions ...
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2answers
78 views

Vectors: difference between $ab$ and $a^Tb$

I have been given a vector problem, np as I am good with vectors. But I was educated in Denmark, and I'm currently in America. The assignment is Find $a^T\cdot b$. Now I have never seen this ...
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1answer
2k views

How to prove the sum of 2 linearly independent vectors is also linearly independent?

Suppose $a,b$ and $c$ are linearly independent vectors in a vector space $V$. How can I prove that $a+b$ or $b+c$ are also linearly independent?
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1answer
2k views

Parametric equation of line parallel to a plane

The parametric equation of the line is $$x=2t+1, y=3t-1,z=t+2$$ The plane it is parallel to is $$x-by+2bz = 6 $$ My approach so far I know that i need to dot the equation of the normal with the ...
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2answers
98 views

Vector cross product and dot product.

Given $a = \langle 1,-1,2\rangle$ and $b = \langle 2,1,0\rangle$ . Find $t$ such that the vector $c = \langle 5,t-1,2\rangle$ is perpendicular to $a \times b$.
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1answer
67 views

Proof that $\operatorname{span}(S_1\cap S_2) \subseteq \operatorname{span}(S_1) \cap \operatorname{span}(S_2)$

An exercise in my book says : Prove that $\operatorname{span}(S_1\cap S_2) \subseteq \operatorname{span}(S_1) \cap \operatorname{span}(S_2)$. Give an example in which $\operatorname{span}(S_1 ...
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3answers
145 views

Possible Cardinality of a Field

The following question struck me as pretty interesting: Let $\Bbb F$ be a field of characteristic $p$ (a prime, of course). I'm then asked to show that $|\mathbb{F}| = p^n$ for some $n\geq 1$. ...
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2answers
63 views

Verifying a bijection

Let $V$ be vector space over $\mathbb{F}$, and $W\subseteq V$ a subspace. Let $p:V\rightarrow V/W$ be the canonical projection. Let $X$ be the set of all subspaces containing $W$ and $Y$ be the sets ...
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2answers
80 views

Proving we have a basis for $F[x]$

So $F$ is an arbitrary field, and $F[x]$ denotes the set of of formal polynomials with coefficients in $F$. And $A=\{f_i \mid i\geq 1\}$. I need to show two things, If $A$ is such that $deg (f_i) ...
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4answers
102 views

Proof that $V = W_1 \oplus W_2$

One of many assignments is: Let $W_1$ and $W_2$ be subspaces of a vector space $V$. Prove that $V$ is the direct sum of $W_1$ and $W_2$ if and oly if each vector in $V$ can be uniquely written as ...
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2answers
581 views

Where is the difference between the union and sum of sets?

My book writes: Definition. A vector space $V$ is called the direct sum of $W_1$ and $W_2$ if $W_1$ and $W_2$ are subspaces of $V$ such that $W_1 \cap W_2=\{0\}$ and $W_1 + W_2 = V$. We denote ...
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2answers
142 views

Proof of finitely dimensional subspaces

Prove: If $V_1, V_2, ...$ are finite dimensional subspaces of a Vector Space $V$, then for $n = 1, 2, ...$ $V_1 +...+ V_n$ is a finite dimensional subspace of V. I have the base case I assume true ...
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1answer
313 views

Basic arc length integration problem

How would you find the arc length of $r(t) = \langle\sqrt{t}, t,t^2\rangle$ for $1\le t\le 4$? This isn't a homework question, I'm just trying to understand how to properly solve a question such as ...
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5answers
167 views

Is closure of convex subset of $X$ is again a convex subset of $X$?

Today I was going through my text book exercise and got hold of the following question. Let $X$ be a normed linear space with norm $\lVert\cdot\rVert$ and $A$ is a non empty convex subset of $X$ ...
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2answers
770 views

Is closure of linear subspace of X is again a linear subspace of X??

Let $X$ be a normed linear space with norm $||\cdot||$ and $A \neq \emptyset$ is a linear subspace of $X$. Prove that $\bar{A}$ is also a linear subspace of $X$.