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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33 views

Find subspaces $W$ and $Y$ of $\mathbb{R}^3$ having the property that $W \cup Y$ is not a subspace of $\mathbb{R}^3$.

I'm prepping myself for graduate linear algebra this fall by attempting self-teach myself some of the "basics" of fields, vectors, etc. found in such linear algebra course. I really don't understand ...
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1answer
26 views

Intersection of planes

A line perpendicular to the plane $ 3x-5y+4z-11=0 $ passes through the origin. At what point does this normal intersects the plane?
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1answer
18 views

Minimal polynomial in primary decomposition theorem

I am going over the proof of the primary decomposition theorem. I can prove that if we have an annihilating polynomial $f$ for some linear transformation $T:V \to V$ and $f$ can be expressed $f = ab$ ...
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1answer
17 views

Heading vector from angle (generated by trig) does not have expected result

I am creating a game and when the player taps on the screen, it should generate a ‘pulse’ effect, pushing away the player. For example, the heading vector should have negative x and y values when to ...
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1answer
35 views

Difference between F-space and Frechet space in W. Rudin's “Functional Analysis”

In Walter Rudin's book, "Functional Analysis", we read that by talking about local base, he will be thinking about neighborhoods of $0$. In the vector space context, the term local base will ...
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1answer
19 views

Is there a function to tell if two probability vectors' max values are in the same dimension?

Is there a method or function to tell two probability vectors' max values are in the same dimension? Or Is there a bound for the angle of two normalized probability vector which their max values are ...
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1answer
40 views

Scalar product and Unit vector

Prove that, for any unit vectors $v_1, v_2, \ldots, v_n$ in $\Bbb R^n$, there exists a unit vector $w$ in $\Bbb R^n$ such that $\langle w, v_i \rangle \leq n^{-1/2}$ for all $i=1, 2, \ldots, n$. (Here ...
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1answer
23 views

problem with invariant subspaces

Consider $V$ unitary space. How to prove that if $T:V\rightarrow V$ and $V\ge U$ is invariant subspace of $T$, then $U^⊥$ is invariant subspace of $T^*$. I know the meaning of invariant subspace ...
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2answers
76 views

Show that no topological vector space is bounded.

I am studying the concept of topological vector spaces in Grubb's Distributions and Operators. A vector space $X$ (over $\mathbb{L} = \mathbb{R}$ or $\mathbb{C}$) is called a topological vector ...
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0answers
20 views

Reformating Function

Is there such a function where a ambiguous ;n-dimensional, field/space (defined by a function) is plugged in and returns a flattened field where the basic units along the function are then formatted ...
1
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1answer
135 views

The intersection of $\operatorname{Im}(T)$ and $\operatorname{Ker}(T)$ is trivial

Let $W$ the space of sequences with entries in $F$ and $S$ the linear operator of $W$ given by: $$S(a_1,a_2,a_3,...) = (a_2,a_3,...)$$ We know that if the intersection of $\operatorname{Im}(T)$ and ...
2
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1answer
49 views

Linear Functional: Continuous? [duplicate]

Given a Banach space: $E$ and chosen a Hamel basis: $\mathcal{B}$ Any vector induces a (noncanonical) algebraic linear functional by: $$\delta:E\to E^*:\delta_b(b'):=\delta_{b,b'}\text{ defined ...
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0answers
18 views

Vectorial product analog operation in 4+ dimensions?

I am thinging about a such operation. Which it need to have: It needs to be $\mathbb{R}^n\times{\mathbb{R}^n}\rightarrow\mathbb{R}^n$ The result needs to be perpendicular to the arguments (thus, ...
2
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1answer
34 views

linearly independent vectors and rows/cols space

Given $n$ vectors, we want to determine if those vectors are linearly independent. One way doing it is writing those vectors as columns of a matrix and row-reduce it. The vectors are linearly ...
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0answers
35 views

Geodesic offset of a curve

Are there standard algorithms that can be easily coded to calculate the offset of a geodesic curve? I am working on computer graphics and it is my first time of working on such algorithms. Please ...
2
votes
3answers
78 views

What is the pushforward of a function (not a vector)

If we have two manifolds $M$, $N$ with the map $f:M \to N$, then this induces a map between their tangent spaces $f_*:T_pM \to T_{f(p)} N$. By duality, another map exists $f^* : T^*_{f(p)}N \to ...
2
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2answers
57 views

When $ax+by+cz+d=0$ is a plane, $a^2 + b^2 + c^2 \neq 0$

I'm reading a book about equation of planes and an way to determinate the equation is to suppose a point $P = (x, y, z)$ And suppose also that $A=(x_0, y_0, z_0)$ is in the plane. $P$ is in the plane ...
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0answers
22 views

Show that $\{w^{1/2}\phi_n\}$ is an orthonormal set in $L^2(D)$ if $\{\phi_n\}$ is an orthonormal set in $L^2_w(D)$

As mentioned in the title, my problem is: Show that $\{w^{1/2}\phi_n\}$ is an orthonormal set in $L^2(D)$ if $\{\phi_n\}$ is an orthonormal set in $L^2_w(D).$ So I know that: ...
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0answers
12 views

Pick a subset of vectors from a set with minimal overlap

This might be more of a comp sci problem, but I was interested more in theory than how to code it. I'm trying to think of a formula/algorithm to do the following: Anyway, so let's say I have a set of ...
0
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1answer
64 views

I want some help of orthogonal vectors

$1.$ Let $u$ and $v$ be orthogonal vectors in $\mathbb{R}^n$ such that $\|u\|=2$ and $\|v\|=3.$ Find $\|2u+3v\|.$ I do it like $\|2u+3v\| < 2\|u\|+3\|v\| = 2.2 + 3.3 = 13$. $2.$ Let $u$ ...
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1answer
21 views

Extenstion of Intermediate Value Theorem.

Let $f:[0,1]^{d}\longrightarrow \mathbb{R}^{d}$ with $d\geq 2$. $f$ is continuous and let $c\in (0,1)$. If we have that $f(0,...,0)<<(c,...,c)$ and $f(1,...,1)>>(c,...,c)$, is there an ...
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7answers
713 views

Why a subspace of a vector space is useful

I'm in a linear algebra class and am having a hard time wrapping my head around what subspaces of a vector space are useful for (among many other things!). My understanding of a vector space is that, ...
3
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1answer
54 views

Vector spaces - $\min\{p\in\mathbb{N}|\text{ker}f^p=\text{ker}f^{p+1}\}=\min\{q\in\mathbb{N}|\text{im}f^q=\text{im}f^{q+1}\}$

$E$ is a $\mathbb{K}$ vector space, $f\in\mathcal{L}_\mathbb{K}(E)$. Let $p\in\mathbb{N}$ so that $\text{ker} f^p=\text{ker}f^{p+1}$ and $q\in\mathbb{N}$ so that $\text{im} f^q=\text{im}f^{q+1}$ ...
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0answers
33 views

Vector spaces question difficulty

I know how to do ALLL the question parts up to (iv) I just don't know how to show the last part (v) . Please help me.
4
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1answer
112 views

Bipartite graph matching like problem.

Let $A=\{a_1,a_2, ..., a_n \}$ and $B=\{b_1,...,b_m\}$ be finite sets. Also $A_1,...,A_k\subset A$ are covering of $A$ and $B_1,...,B_t\subset B$ are covering of $B$. Let $V$ be a set of pairs of ...
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2answers
49 views

Angle between two vectors on manifold

I'm parallel transporting a vector along a curve and trying to calculate how much this vector rotates relative to the curve's tangent vector. So if the path is a geodesic then I will get an answer of ...
3
votes
2answers
126 views

Relationship between the singular value decomposition (SVD) and the principal component analysis (PCA). A radical result(?)

I was wondering if I could get a mathematical description of the relationship between the singular value decomposition (SVD) and the principal component analysis (PCA). To be more specific I have ...
1
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1answer
33 views

Elementary problem about Tensor product and Kronecker product defined by linear map

I have some perplexities when I reading references about tensor product. My main question is: How to define the tensor product between two vectors? It is clearly to define the tensor product ...
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1answer
33 views

Proof that Vector Space in Domain of Linear Map is a Direct Sum

I'm working through problems in Linear Algebra just for fun and I am getting stuck on Axler 3.4. Suppose that $T$ is a linear map from $V$ to $\mathbf{F}$. Prove that if $u \in V$ is not in $null\ ...
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1answer
46 views

How to interperet calculus thing

I have $\nabla \times (f\mathbb{F})$ where $f$ is a twice continuously differentiable scalar field and $\mathbb{F}$ is a twice continuously differentiable vector field. Is it right to interpret $f$ ...
3
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0answers
41 views

Vector Space Verification

I just took an exam asking me if the following are a vector space over $\mathbb{R}$ assuming that the set of all real valued functions on the interval $[0,1]$ is a vector space with theoperations ...
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1answer
19 views

Show that $\mathcal{C}(A)$ is the smallest convex of $E$ containing $A$.

Let $E$ be a $\mathbb{R}$-vector space, and $A$ a nonempty subset of $E$. Show that $$\mathcal{C}(A) = \biggl\{\sum \limits_{k=1}^n \lambda_kx_k \biggm| n \in \mathbb{N}^*, (x_1,\dots,x_n) \in ...
2
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3answers
57 views

Let $V $be a vector space. Prove/Disprove: There is a norm $\|\cdot\|$, such that all subsets of $V$ are open sets in $(V,\|\cdot\|)$.

The Assignment: Let $V$ be a vector space over $\mathbb{R}$ with $V \not= \{0\}$. Prove or disprove: There is a norm $\|\cdot\|_d$ on $V$, such that all subsets of $V$ are open sets in ...
2
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1answer
31 views

Why do special solutions of $Ax=0$ form a basis for null-space of $A$?

I read somewhere that The $n-r$ special solutions of a $m \times n$ matrix with rank $r$ form a basis for its null-space. If we consider the general RREF for the given matrix, it has the form: ...
0
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1answer
29 views

linear space proof

The question is that V is the span of these vectors in the diagram b2,b3,b4. Please help me in this problem, I know all the theory that for it to be a linear space it should be closed under addition ...
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3answers
34 views

Definition of linear independence when $v_i=0$

I have read that linear independence occurs when: $$\sum_{i=1}^n a_i v_i =0$$ Has only $a_i=0$ as a solution, but what if all $v_i$ were $0$ then $a_i$ could vary and still yield $0$. Does that mean ...
0
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1answer
24 views

finding the Rank and basis of null space of this matrix

Please help me with this question. The question is to find the rank of the matrix and then the basis of the null space, I first put the matrix A in reduced row echelon form and then I wrote the ...
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1answer
85 views

How to prove $ |\langle u,v\rangle| \leq ||u||||v||$

How to prove $ |\langle u,v\rangle | \leq ||u||||v||$ Note: I have given this many attempts so don't downvote due to lack of effort, refer to edit history for evidence of said effort
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3answers
152 views

Subspace of a finite dimensional inner product space, independence of basis choice

Let $W$ denote a subspace of a finite dimensional inner product space $V$, and let $$\beta = \{w_1,w_2,\dots,w_r\}$$ denote an orthogonal basis for $W$. For any $v\in V$ define $$proj_{\beta}v = ...
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0answers
27 views

Proving that combinations of these linear spaces are not linear spaces [duplicate]

The vectors $\mathbf{b}_1$, $\mathbf{b}_2$, $\mathbf{b}_3$, $\mathbf{b}_4$ are defined as follows: $$ \mathbf{b}_1 = \begin{pmatrix}1\\0\\0\\0\end{pmatrix}, \mathbf{b}_2 = ...
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2answers
68 views

Linear Space in Vector Spaces question.

How do I do the first part of the question where they say V1 U V2 is not a linear space, please help my exam is very close. In the marking scheme it says it's not closed under addition but can ...
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0answers
15 views

Sufficient conditions for RTree

What is the sufficient screening criteria of a space for the possibility to use R-Tree spatial index on it? I cannot apply it to a space with just Jaccard distance as the metric. As I suppose the ...
3
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1answer
22 views

Is there any example of usage for a vector space over the field of formal Laurent series?

The formal Laurent series over a field is a field. Is there any example where vector spaces over that field occur naturally?
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1answer
12 views

Calculating a basis in $\mathbb{R}^4$.

Have the subspace of $\mathbb{R}^4$ $$W = \{(x,y,z,w) \in\mathbb{R}^4 : y - w + z = 0\}$$ Calculate a basis for $W$, and then find an orthonormal base from that. The basis, from the ...
0
votes
1answer
24 views

How does full row rank imply column space is $R^m$ for a $m \times n$ matrix?

From Gilbert Strang's textbook Introduction to Linear Algebra (p.159) Every matrix with full row rank has these properties $Ax=b$ has a solution for every right side $b$. The column ...
0
votes
1answer
32 views

Calculating a basis given some constraints.

Have a vector space formed by the vectors $(x_1,x_2,x_3,x_4)$ that satisfy $$\begin{cases} x_1+x_2-x_3-3x_4=0\\ 2x_1+x_3-2x_4=0 \end{cases}$$ Find a basis and also the dimension of $S$. ...
2
votes
1answer
28 views

Does linear dependency have anything to do when determining a span?

Q: Does $\{(1,1) , (2,2)\}$ span $\mathbb{R}^2$? A: No, because they are linearly dependent. I agree that it doesn't span $\mathbb{R}^2$, but from my understanding, linear dependency has ...
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3answers
23 views

Proving that a subset is a subspace by showing a scalar combination.

Prove that: $$S = \left\{\left(\begin{matrix}a & b \\ c & a\end{matrix}\right) \ / \ a,b,c \in \mathbb{R}\right\} \subset M(2,\mathbb{R})$$ Answer: $S$ is a scalar ...
1
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1answer
41 views

When proving if a subset is a subspace, can I prove closure under addition and multiplication in a single proof?

I'm learning about proving whether a subset of a vector space is a subspace. It is my understanding that to be a subspace this subset must: Have the $0$ vector. Be closed under addition (add two ...
1
vote
1answer
35 views

Using rgb triplets in “scalar” math

So I'm reading a computer graphics research paper and I'm confused as to how to interpret certain formulas that are being used. In the paper, a value σ is defined as an RGB triplet. Later on, that ...