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

1) $Im(g ◦ f) ⊆ Im(g) ∧ Ker(f) ⊆ Ker(g ◦ f)$ 2) $rg(g ◦ f) ≤ min(rg(g), rg(f))$ 3) $dim(Ker(g ◦ f)) ≤ dim(Ker(f)) + dim(Ker(g))$ I'm lost with this question. Where and how do I start? Thank you in ...
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How are Fn and Finfinity special cases of Fs?

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
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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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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
41 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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36 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
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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
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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
25 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
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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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If $a^2 = b^2$ then which values $a$ and $b$ are constrained to be? [on hold]

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
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Subspace generated [on hold]

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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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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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 ...
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2answers
58 views

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

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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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
31 views

derivative in a vector direction [on hold]

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
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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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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
30 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
55 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
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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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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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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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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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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
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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
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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
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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 ...
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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 ...
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1answer
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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 ...
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1answer
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Finding algebraic expression of a parallepiped given directions and side lengths

Let $u_1,\dots,u_n$ be linearly independent unit vectors in $\mathbb R^n$. Let $T$ be the parallepiped centred at $0$, with sides parallel to $u_i$ and side lengths $l_i$, $i=1,2,\dots,n$. My question ...
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1answer
22 views

Extend basis of a subspace to a basis of $\mathbb{C}^5$

Let $U$ be the subspace of $\mathbb{C}^5$ defined by $$U = \{(z_1, z_2, z_3, z_4, z_5) \in \mathbb{C}^5: 6z_1 = z_2\;\wedge\; z_3 + 2z_4 + 3z_5 =0 \}$$ (a) Find a basis of $U$ (b) Extend ...
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Gradients on a hyperplane

Consider the hyperplane $H:=\lbrace x\in\mathbb{R}^{n}\ \colon\ x_{1}+\dots+x_{n}=0\rbrace$. If we treat $H$ as a manifold, and $f\colon H\to\mathbb{R}$ is smooth, then what is the gradient $\nabla f$...
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What is the dimension of the Four Fundamental Subspaces of a $2\times 2$ matrix A with rank $r(A)=1$?

I am taking the 18.06 Linear Algebra course on MIT OpenCourseWare, and I came across problem 30 on section 3.6 of professor Gilbert Strang's book (Introduction to Linear Algebra). The problem states:...
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2answers
55 views

How to determine the dimension and a base of U, V, U ∩ V, U + V?

We have $\Bbb R^3$ with the following linear subspaces: $U = \{(x_1, x_2, x_3) \in \Bbb R^3 |~x_1 + x_2 − x_3 = 0\}$ $V = \{(x_1, x_2, x_3) \in \Bbb R^3 |~2x_1 − x_2 + x_3 = 0\}$ In each case I ...
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3answers
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Interesting subspace of $M_n(\mathbb{C})$ [CMI 2019]

Cosider the vector space $V=M_n(\mathbb C)$ and $W$ be a vector subspace of $V$ such that every non-zero element of $W$ is invertible. Show that dim$W=1$. I know that any matrix over $\mathbb C$ is ...
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1answer
31 views

Decide a number $a, b, c$ (expressed by $\lambda$) so that $a\mathbf{u}_1 + b\mathbf{u}_2 + c\mathbf{u}_3 = \mathbf{x}$

See the following vectors in $\mathbb{R}^3$: $ \mathbf{u}_1= \begin{pmatrix} 1 \\ -1\\ 1 \\ 1 \end{pmatrix} $, $ \mathbf{u}_2 = \begin{pmatrix} 1 \\ 2 \\ 0 \\ 2 \end{pmatrix} $, $ \mathbf{...
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2answers
26 views

Decide whether the following subsets of a vector space are sub-vector spaces.

a) $\{f\in\mathbb{R}[t]:f(1)=0\}=:U_1$ b) $\{f\in\mathbb{R}[t]:\exists a\in\mathbb{R}\text{ with }f(a)=0\}=:U_2$ where $\mathbb{R}[t]$ is the set of all polynomials above K. Does a) mean, that ...
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2answers
32 views

Are $W_1$ and $W_2$ same? How they are different?

Let \begin{align} W_1&=\{(a_1,a_2,...,a_n)\in \mathbb C^n \mid a_1=\mu a_n, \text{for some fixed $\mu \in \mathbb C$} \}\\ W_2&=\{(a_1,a_2,...,a_n)\in \mathbb C^n \mid \frac{a_1}{a_2}=\mu , \...
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2answers
41 views

Determine the dimension $\dim$ and a basis of $U_1$,$U_2$, $U_1+U_2$ and $U_1\cap U_2$ in each case.

The sub-vector spaces $U_1=span\{(0,1,2),\;(1,1,1),\;(3,5,7)\}$ and $U_2=span\{(1,1,0),\;(-1,2,2),\;(2,-13,-10),\;(2,-1,-2)\}$ of the $\mathbb{R}^3$ are given. Determine the dimension $\dim$ and a ...
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1answer
47 views

How to specify a base for each of the following vector spaces and prove the claims?

Specify a base for each of the following vector spaces and prove your claims: a) $\{(x_1,x_2,x_3)\in \mathbb{R}^3:x_1=x_3\}$ over $\mathbb{R}^3$ b) $\{f=\sum^n_{i=0}a_it^i\in \mathbb{Z}_7[t]:...
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0answers
19 views

Is the expectation of the inner product of two random vectors the inner product of the expectation of each one individually?

Am trying to figure this out. Think the answer might depend on the specific spaces on operators involved, but any help you could give would be much appreciated!