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Questions tagged [vector-spaces]

For questions about vector spaces and their properties. More general questions about linear algebra belong under the [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 where we can make sense of linear combinations.

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26 views

Find basis to sub vector spaces

V = $\Bbb R_3 [x]$ and W,U $\subseteq$ V are sub vector spaces. U=$span${$1 - x, x^2, x^2-x^3, -1+x-x^2+2x^3$} W={p(x)$\in\Bbb R_3 [x]$ | p(1)=0 ^ p(2)+p(0)=0} Find basis to W, U+W, U$\cap$W
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1answer
22 views

Self-adjoint operator is diagonalisable

I am revising adjoints for a linear algebra exam and am confused as to how to prove this. Suppose that $T: V \rightarrow V$ has the property that $T^*=aT$ for some complex a. How then do you prove ...
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0answers
20 views

Basis to vector space with all inverse matrices

$V$ is sub vector space of $M$(2x2) $(\Bbb R)$, V= { $\begin{matrix}a+b & a+e & \\ c-2d&4c-8d\\\end{matrix}$} when $a,b,c,d,e \in \Bbb R$. Give an example of basis to V, that ...
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0answers
28 views

What makes a vector superfluous? [on hold]

In Howard Anton's Elementary Linear Algebra -> section 4.3 Linear Independence -> it is said that if we introduce a third coordinate axis in rectangular coordinate system such that the third axis ...
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3answers
23 views

Do the following sets span $P_{3}(\mathbb{R})$?

Could somebody please confirm, if I have done this right or whether my approach is right? Consider a $\mathbb{R}$ -Vectorspace $P_{3}(\mathbb{R})$ of real polynomials: $a_{0}+a_{1} X+a_{2} X^{2}$ ...
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0answers
23 views

Confusion on the Gram-Schmidt process for complex vectors

I am having some trouble with the inner product and the Gram-Schmidt process for complex vectors as I am trying to learn it on my own. This is mainly due to the discrepancy with my text book and what ...
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0answers
32 views

In what sense is the column space of a symmetric matrix equivalent to its row space?

I have an elementary question about linear dependence and column/row spaces. Suppose we have a matrix $A=\begin{bmatrix}1&2\\2&4\end{bmatrix}$, and we construct the matrix $B=\begin{bmatrix}1&...
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0answers
9 views

Derivation of Hessian of a log likelihood function

I have to find the Hessian of log likelihood function, i.e, $E\{-\frac{\partial \log p(x_{k+1|k})}{\partial x_k \partial x_k^T }\}$, where $p(x_{k+1|k}) \sim N(f_k(x_k), Q)$. Here $x_k \ \& \ ...
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1answer
56 views

How can this set be a vector space? [on hold]

How can the set {x, y, 0} be a vector space if x+y is not an element of the set? Edit: I mean where x and y are both elements of F.
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1answer
29 views

Proof verification for linear subspaces from Michael Taylor's linear algebra notes

Let $V$ be a vector space over a field $\mathbb{F}$ and $W,X\subset V$ linear subspaces. We say $$V=W+X$$ provided each $v\in V$ can be written $$v=w+x,w\in W,x\in X.$$ We say $$V=W\oplus X$$ ...
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1answer
30 views

Hamel dimension of a vector space, and dimension of the dual

I have the following (possibly trivial) observation: Let $K$ be an $\mathbb{F}$-vector space (I believe the argument also works for free modules), and let $X\subseteq K$ be it's basis with ...
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1answer
35 views

Vector spaces are not rigid

Im following Etingofs Tensor categories and have read about rigid categories now. There it says The category of all vector spaces (including infinite dimensional) is not rigid. Take $V$ to be ...
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0answers
24 views

Let $A, B, C$ be finite-dimensional K vector spaces and $f: A → B$ and $g: B → C$ linear mappings. Show the following:

1) $\mathrm{Im}(g ◦ f) ⊆ \mathrm{Im}(g) ∧ \mathrm{Ker}(f) ⊆ \mathrm{Ker}(g ◦ f)$ 2) $rg(g ◦ f) ≤ \min(rg(g), rg(f))$ 3) $\dim(\mathrm{Ker}(g ◦ f)) ≤ \dim(\mathrm{Ker}(f)) + \dim(\mathrm{Ker}(g))$ I'...
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0answers
22 views

How are Fn and Finfinity special cases of Fs? [on hold]

If Fn represents the set of all lists of length "n" and Finfinity represents the set of all infinitely long lists, how are Fn and Finfinity special cases of Fs? Here if S is a set, Fs represents the ...
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1answer
17 views

Why does the union of $k$-dimensional subspaces contain no „new“ $k$-dimensional subspaces?

For simplification, let $V$ be a finite-dimensional, real vector space. I know that I cannot represent $V$ as the union of finitely many proper subspaces (or even just $k-1$-dimensional subspaces). ...
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0answers
51 views

Are all isometries of subsets affine?

I've found at least two questions that deal with whether isometries are affine, Are isometries always linear? and Should isometries be linear? However, both of these questions assume we are ...
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1answer
43 views

Is My Understanding Correct

I know there's already some similar questions on this site, but I wanted to phrase my question in a slightly different way. My question is What does "$F^n$ is a vector space over $F$" mean? I ...
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0answers
38 views

“vector spaces” without additive inverses

Setup: I know the definition of a vector space: a set $V$ over a field $F$ such that is closed under vector addition and scalar multiplication. My question: Is there a name for spaces that are ...
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4answers
33 views

How to do the curl of the product of a function and a vector field

I have the following property which says: $∇ × (f G) = ∇f × G + f (∇ × G)$ Where $f$ is a differentiable function in an open set $S$ on $\mathbb {R}^3$ , and $G$ is a $C^1$ class vector field on $S$ ...
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1answer
24 views

Total Variation of Subintervals

Studying functions of bounded variation, the following exercise showed up: Let $I = [a,b]$ be an interval, $(E, \| \cdot\|)_E$ a normed vector space and $f \in BV(I, E)$ a function which is ...
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2answers
28 views

Linear Algebra Scalar and Vector Projection [on hold]

I have a problem with one part of a question. I just need help with part b. How would I go about applying the equation to find the scalar and vector projection of v2 onto v1? Thanks!
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1answer
53 views

radius of the inscribed sphere of the pyramid in $\mathbb{R^5}$

How to find the radius of the inscribed sphere of the pyramid in $\mathbb{R^5}$ with the vertex in $(1,0,0,0,0)$, which base is a regular $4$-dimensional simplex, luying in the hyperplane $x_1=0$ with ...
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2answers
39 views

If $a^2 = b^2$ then which values $a$ and $b$ are constrained to be? [closed]

I've the following subset of $\mathbb{R}^3$: $$ Y= \{(a, b,c)^T | a^2=b^2\} \subset \mathbb{R}^3 $$ How can I embed the condition $a^2=b^2$ into the vector? That is, what can I say about $a$ and $b$?...
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2answers
14 views

Subspace generated [closed]

Let V be the space of the matrices 2 x 2 on R, and let W be the subspace generated by: Find a base, and the dimension of W. Anyone can help me with this problem?
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0answers
61 views

Does the convex cone of monotonic functions on a compact set admit a countable conic generating set?

For any function $f:[0,1]^d\rightarrow\mathbb{R}$ and any $i\in\{1,\dots,d\}$ and $x\in[0,1]^d$, let the functions $f_{i,x}:[0,1]\rightarrow\mathbb{R}$ and $\partial_i f:[0,1]^d\rightarrow\mathbb{R}$ ...
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1answer
25 views

Light Reflection in a 3D Plane [Raytracing]

This is a problem that I need to figure out to complete a Java Raytracer Render Engine. Let's say a Light is positioned at (x, y, z) position in a 3D Environment. There's also a metallic object, let'...
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0answers
37 views

Choosing vectors for projection

I have a vector $v = a_1\mu_1 + a_2\mu_2 + ... + a_n\mu_n$ where $\mu_i$ are given linearly independent but not orthogonal vectors. I need to choose $k$ vectors from the original set such that when ...
3
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2answers
59 views

Prove that the equation $ax = b$, $xa = b$, always has a unique solution. [closed]

A vector space $A$ is called an algebra if in it, in addition to the addition of vectors and multiplication by a number, the multiplication of vectors with properties is defined. In other words, for $...
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1answer
17 views

A Vector Divided by it's Distance to a Subspace Yields a Vector with Distance 1 to Subspace

Let X be a normed vector space, S a subspace of X and x∈X. Distance is defined by: $$|x,S|:=\inf||x-s||, x\in X, s\in S$$ How does one prove that for: $$z:=\frac{x}{|x,S|}\Rightarrow |z,S| = 1$$ This ...
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0answers
26 views

I attempted to visualize dot product of complex vectors. What do you advice?

I am an electronics undergraduate student currently learning wavelets. In the book A First Course in Wavelets with Fourier Analysis authors first introduce complex vectors and their dot products. Then ...
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1answer
32 views

derivative in a vector direction [closed]

Could anyone please help me with this question? I'm not sure that I understand how to get the derivative of the direction of a vector. The derivative of the function $f(x,y) = 2xy^3 – 3x^2y$ at the ...
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2answers
50 views

How to determine the base of $\ker\phi$ for polynomial function?

Given is a base defined as $$B:=(x\mapsto1,x\mapsto x,x\mapsto x^2,x\mapsto x^3 ,x\mapsto x^4)$$ A set V defined as $$V:= \{ f: \mathbb{R} \mapsto \mathbb{R}\ |\ \exists\ {a_0},...{a_4} \in \mathbb{R}\...
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1answer
26 views

How to calculate a point after rotation given two unit vectors?

I have two unit vectors: before and after rotation. Point (0, 0, 1) is moved to (-0.42, 0.19, 0.88) after rotation. If I had a point of (-0.066, 0.635, -0.184) before rotation, how it would be ...
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0answers
15 views

Checking if lines intersect or skew from directions ratios and a point on each line

Well , this is what's in my book $DR_1(m1,m2,m3)$ with $A(a1,a2,a3)$ $DR_2(n1,n2,n3)$ with $B(b1,b2,b3)$ \begin{vmatrix} m1 & m2 & m3 \\ n1 & n2 & n3 \\ a_1-b_1& a_2-b_2 & ...
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1answer
20 views

Systematic approach to find base of vectorspace given its elements' traits

I'm trying to find a base for a vector space that's given as a set with certain traits. Take this example: Let $V$ be an $\mathbb{R}$-vector space with $$ V := \left\{ (a, b, c, d) \in \mathbb{R}^4 : ...
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3answers
33 views

How to find basis of linear subspace $V$ when $V$ contains all polynomials with the degree up to 4?

I have been given the following definition $V := \{ f : \mathbb{R} \to \mathbb{R} \mid \textrm{ there are } a_0,\ldots,a_4 \in \mathbb{R}$ and $f(x) = \sum_{i=0}^4 a_i x^i$ for all $x \in \mathbb{R}\}$...
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1answer
20 views

Basis of line in vector space

I puzzled by the following question I found in Linear Algebra and its Applications by Gilbert Strang: Find a counterexample to the following statement: If $v_1$ , $v_2$ , $v_3$, $v_4$ is a basis ...
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1answer
27 views

Number of all subspaces of an $n$ dimensional vector space over $\mathbb{Z}_2$ [duplicate]

What is the best known lower bound for the number of all subspaces of an $n$ dimensional vector space over $\mathbb{Z}_2$? P.S: For the record, I am aware of Gaussian Coefficients, i.e., the number ...
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3answers
56 views

Prove angle addition holds for $\mathbb{R}^n$

Define $\theta(u,v)=\cos^{-1}(\frac{u\cdot v}{|u||v|})$ be the angle between $u,v\in \mathbb{R}^n$, where $u\cdot v$ is the standard inner product and $|x|=\sqrt{x\cdot x}$ for all $x\in \mathbb{R}^n$....
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2answers
24 views

How to determine if vectors are generators of a vector space.

How can I determine if vectors $\vec{v}$ of $B=\left \{\begin{bmatrix}1 & 0 \\ 1 & 1\end{bmatrix}, \begin{bmatrix}-1 & 0 \\ 0 & 1\end{bmatrix}, \begin{bmatrix}0 & 1 \\ 1 & 1\...
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1answer
35 views

Orthogonal projection of the vector $p=(1,0,0,0)$ onto the subspace $W=[(1,-3,0,1),(1,5,2,3),(0,4,1,1),(1,-2,0,4)]$

I got a subspace $W=[(1,-3,0,1),(1,5,2,3),(0,4,1,1),(1,-2,0,4)]$ and I want to make an orthogonal projection of a vector $p=(1,0,0,0)$ onto $W$ and onto the orhhogonal complement of $W$. ...
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0answers
26 views

Proof that the vector space of continuous functions from $[0,1]$ to $\Bbb R$ is infinite-dimensional

I want to use the following lemma: A vector space V is infinite-dimensional if and only if there exists a sequence of vectors $v_1,v_2,...$ such that $v_1,v_2,...,v_n$ is linearly independent for ...
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0answers
21 views

Explicit description of a column space

In a task I should obtain the explicit description for ColA of a given matrix A. The problem that I have is, that I don't know what explicit means in this context. Do I have to write down the columns ...
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0answers
11 views

plane edge intersections embedded in higher dimensional space

Let's say we have some D-dimensional euclidean space, and we have some circles of dimension 0 to D-1 (circle dimensionality meaning the minimum number of vectors needed to fully define it, so a ...
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0answers
21 views

Dimension of a vector space with addition and scalar multiplication

The dimension of a vector space is determined by the underlying field $\Bbb{F}$, and the addition and scalar multiplication operations (given the set of vectors $V$). I am aware of how changing the $\...
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5answers
68 views

What is $\dim(V^V)$? [closed]

Where $V$ is a vector space and $V^V=\left\{ f\mid f\colon V\rightarrow V \right\}$. I proved that $V^V$ is a vector space so $\dim\left(V^V\right)$ is well defined.
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2answers
33 views

Can this exponential be complex valued?

My complex analysis is very sketchy, and I am a little stumped by the following - although it seems incredibly innocuous. For $t\in\mathbb R$ and a fixed parameter $\alpha\in\mathbb R/\{0\}$ does it ...
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1answer
19 views

Show that the set of LES-solutions form a sub-vector space of $\mathbb{R^n}$ exactly when $b_i = 0$

The linear system of equations is given: \begin{align} a_{11}x_1+\dots& +a_{1n}x_n=b_1\\ &\vdots\\ a_{m1}x_1+\dots&+a_{mn}x_n=b_m \end{align} Show that the set of the given linear ...
0
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0answers
28 views

N-Dimensional Sphere intersections embedded in higher dimensional space

Let's say we have some D dimensional Euclidean space. Let me use the term S-Sphere to only indicate spheres that match the dimensionality of the space they reside in, while Circles are spheres with ...
0
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1answer
32 views

Show that $V_1 \cup V_2 = V$ implies $V_1 = V $ or $V_2 = V$ [duplicate]

I'm troubled with the following question: $V_1$ and $V_2$ are subspaces of $K$-vector space $V$. Show that $V_1 \cup V_2 = V$ implies $V_1 = V$ or $V_2 = V$. I don't understand why $V_1$ or $V_2$ is ...