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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How to show that the surjectivity of a linear map $f: R^ n\to R^n$ implies the injectivity and vise versa?

How to show that the surjectivity of a linear map $f: R^ n\to R^n$ implies the injectivity and vise versa?
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1answer
17 views

On Equivalent Norms in an Infinite Dimensional Vector Space

How many non-equivalent norms can we define in an infinite dimensional vector space? Is there any explicit expression?
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1answer
18 views

By given equation, finding orthogonal projection

Find the orthogonal projection of line with direction vector $u = ( 1 , 2 , 0 )$ onto the plane described by equation $-3x - 2y + 2z = -2$ i have tried to search ...
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3answers
27 views

How to determine volume of parallelepiped by 4 points

Let points $(0,0,0), (1,2,x), (-2,1,0)$ and $(1,1,3)$ be at four corners of a parallelpiped. Determine the volume of the parallelepiped by using the determinant in terms of $x$. For what value of $x$ ...
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3answers
54 views

Does there exists a vector v such that $Av\neq 0$ but $A^{2}v=0$

Let A be a $4\times4$ matrix over C such that $\operatorname{rank}A=2$ and $A^{3}=A^{2}\neq0$. Suppose that A is not diagonalizable. My question is , "Does there exists a vector $v$ such that ...
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32 views

Prove that there is a subset with an invertible sum

The question is as follows: For some $m>n$, let $A_1,\cdots,A_m$ be $n\times n$ matrices, satisfying $$ A_1+\cdots+A_m=I_n $$ where $I_n$ is the $n\times n$ identity matrix. Show that ...
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1answer
11 views

Cauchy sequence of vectors when dotted with another vector gives a Cauchy sequence of scalars?

My question is related to vector spaces with an inner product defined (the space is not necessarily complete i.e. not a Hilbert Space) So imagine I have a Cauchy sequence of vectors ...
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2answers
20 views

Linear Algebra proof for projections

How would i solve the proof proja(proja(b)) = proja(b) I subbed in the projection formula of ((a dot b)/(abs(a^2)) x a but did not get the answer
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0answers
19 views

plane generated by n linearly independent n-dimensional vectors

Prove that the following statement is true. I'm not sure whether the term 'linear combination (narrow sense)' is widely used since I'm studying in Korea. According to my professor, the term ...
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2answers
28 views

prove cauchy-schwarz inequality (missing a step)

during lecture notes I only caught most of the proof and couldnt write a step down fast enough, and I'm having a touch trouble seeing how to get from the previous step to the next. Here is what i have ...
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1answer
25 views

Transpose of composition of functions

I am trying to find an alternative proof that $(AB)^t=B^tA^t$. I think it is possible to do by showing that: $(g \circ f)^t=f^t \circ g^t$ where f,g are linear maps between vector spaces. I am ...
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1answer
12 views

Association of a vector space to metric, normed and inner product spaces

There is a nice visual representation of mathematical spaces from this post: I am not quite sure how vector spaces fit into this image. I know metric space is not necessarily a vector spaces, but ...
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1answer
35 views

If A^2 =0 then possible rank of A

Let, A be a non zero matrix of order 8 with A^2 =0. Then one of the possible value for rank of A is (a) 5 (b) 4 (c) 6 (d) 8
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7 views

Number of pivot columns in a 4x6 matrix for spanning set to occur

How many pivot columns must a 4x6 matrix have if its columns span $\mathbf{R}^4$? Explain. So, in my head, this is pretty clear: You need four dimensions => So you need a minimum of four vectors that ...
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1answer
18 views

Calculate projection of a line in a square

Said that we have two points (P1, P2) that form a line, and 3 points (S1,S2,S3) that form a square, how would we calculate the position X and Y of the point resulting from the intersection of the line ...
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0answers
13 views

Subspace vector proofs problem [on hold]

I'm having trouble understanding/solving this proof. QUESTION: Prove the set P_3 is a subspace of P_4 with standard operations, where P_n is a vector space of all polynomial functions with degree n ...
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0answers
17 views

Derivative of vectors dimension do not agree

I have two n by 1 vectors $\mathbf w,\mathbf v$ with respect to $\mathbf w$, and $\mathbf v$ is some function to $\mathbf w$. so I can get a scalar from $\mathbf w^T\mathbf v$, I want to take ...
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0answers
9 views

Vector finding midpoint

The points P and Q have position vectors, relative to the origin O, given by $OP = 7i + 7j - 5k $ $ OQ= -5i + j + k$ The mid-point of PQ is the point A How to find A ? I try to find the vector ...
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1answer
17 views

Which of these sums is equal to $\mathbb R^4$?

I'm given the following sets: $$U=\{(0,a,b,a-b): a,b \in \mathbb{R}\} \\ V=\{(x,y,z,w): x=y, z=w\} \\ W=\{(x,y,z,w): x=y\}$$ I'm trying to determine which of the following is equal to $\mathbb R^4$: ...
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0answers
20 views

How to show that the vertices of a convex hull are given by these specific subsets…

We work over $\mathbb{R}^N$. Let $V$ be the corners of the unit cube $[0,1]^N$, or equivalently the set of vectors whose coordinates take values $0$ or $1$. Let $d:\{0, \ldots, N\} \to \mathbb{R}_+$ ...
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1answer
16 views

Rank of a linear transformation T

Let, n be a positive integer & let M_n(R) be the space of all n*n real matrices. If T:M_n(R)-->M_n(R) is a linear transformation such that T(A)=0, whenever A belongs to M_n(R) is symmetric or ...
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2answers
20 views

Dimension of matrices with entries $a_{ij} = a_{rs}$ with $i+j = r+s$.

Let $n$ be a positive integer and $H_n$ be the space of all $n \times n$ matrices $A = (a_{ij})$ with entries in $\Bbb{R}$ satisfying $a_{ij} = a_{rs}$ whenever $i+j = r+s \; (i,j,r,s = 1, 2, \ldots, ...
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0answers
13 views

How to understand affine space and affine transformation

As far as I know, the affine space is a space without origin point. Some others define affine space as $$A=\{\sum_{i=1}^N \alpha_i \boldsymbol{v_i|\sum_{i=1}^N}\alpha_i=1\}$$ How do we relate these ...
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1answer
24 views

Do the given vectors span $\mathbb{R}^3$?

Do the following vectors span $\mathbb{R}^3$: $$v_1 = (2, -1,3)$$ $$v_2 = (4, 1, 2)$$ $$v_3 = (8, -1, 8)$$ I use Gaussian Elimination to bring the matrix to an echelon form, with a pivot of "1" in ...
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0answers
21 views

Can a Norm be Induced by two Different Complex Inner Products?

Let $(X,\|\cdot\|)$ be a normed vector space over $\mathbb{C}$. If $\|x\|=\sqrt{\langle x,x\rangle}$ and $\|x\|=\sqrt{\langle x,x\rangle'}$ for all $x\in X$ where $\langle,\rangle$ and ...
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2answers
31 views

Show that a unique matrix exists for the coordinate vectors in a vector space

If $A=\{a_1,...,a_n\}$ and $B=\{b_1,...,b_n\}$ are two bases of a vector space $V$, there exists a unique matrix $M$ such that for any $f\in V$, $[f]_A=M[f]_B$. My textbook uses this theorem ...
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1answer
40 views

Proofs for $n$-dimensional vector spaces $V$

Suppose $V$ is an $n$-dimensional vector space. Prove that there is at most $n$ linearly independent elements in $V$. Prove that a set of $m<n$ element in $V$ cannot span $V$. I'm not really ...
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1answer
27 views

Find the N versors more 'spaced' [on hold]

I have to deal with a concrete problem that is: Given a 3d object I want to select N directions with N integer and N>=3 for projection that would maximize the information I gain and thus my ability to ...
2
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0answers
38 views

Regarding Linear Subspaces over a Finite Field… TFAE:

Let $V=\mathbb{F}^n$, for a finite field $\mathbb{F}$. Prove the equivalence of the following statements: There is a linear subspace $C$ of $V$ with the property that every vector $v$ of ...
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8 views

What is meant by $\Omega=\text{cls}\left\{a_{\tau}|\tau\in\mathbb{R}\right\}$?

Let $\mathcal{B}=\left\{b\colon\mathbb{R}\to M^n| b \text{ is uniformly bounded and uniformly continious}\right\}$; give $\mathcal{B}$ the compact-open topology. It can be shown that the map ...
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1answer
19 views

Continious subbundle

Let $W$ be a vector bundle with base $\Omega$ and projection $p$. A continious subbundle of $W$ is a subset $W_0$ of $W$ such that $p|W_0$ defines a vector bundle over $\Omega$. Now here ...
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0answers
10 views

dim of $\Bbb R^3 \otimes_\Bbb R \Bbb C$ when considering as a $\Bbb C$-vector space

I'm looking at Sergei Winitzki's Linear Algebra via Exterior Products, and he has a question on tensor products. Firstly we construct the real vector space $\Bbb R^3 \otimes_\Bbb R \Bbb C$ which is ...
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1answer
17 views

Reducing Subspaces: Nonexample?

Given a Hilbert space $\mathcal{H}$. Consider an operator $T:\mathcal{D}(T)\to\mathcal{H}$. Suppose there exists a closed subspace $Z\leq\mathcal{H}$: $$TZ\subseteq Z,TZ^\perp\subseteq Z^\perp$$ ...
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23 views

Problem in proving norm

I have a linear space V that includes the continuous functions from [-$\pi, \pi$] to the complex set C, of the form: $$f(x) = a\cos(t)+b\sin(t) $$ where $a$ and $b$ are complex numbers. I want to ...
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1answer
22 views

Vector spaces and direct sums

The map that was constructed in lectures is: $V,W$ subspaces of $U$. $f\colon V \oplus W \to U$ by the formula: $f((v,w))=v+w$ for $v$ in $V$, $w$ in $W$ Is it correct to generalise this to, ...
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2answers
34 views

Is union of two subspace a subspace too? [duplicate]

Assume that $W$ and $V$ are two subspace of $X$. Is their union a subspace of $X$ too? I think it is not true unless under certain conditions but I do not know what conditions...
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1answer
36 views

Proof vector x + ⃗y = ⃗x + ⃗z then ⃗y = ⃗z

Let ⃗x, ⃗y and ⃗z be vectors in a vector space V . Prove that if ⃗x + ⃗y = ⃗x + ⃗z then ⃗y = ⃗z. No idea how to start.
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1answer
19 views

Prove that the linear map of the basis $V$ is a spanning set of the image of $f$

Suppose that $f:V\rightarrow W$ is a linear map of finite-dimensional vector spaces and that $S=\{v_1,v_2,...,v_n\}$ is a basis for $V$. Prove that $\{f(v_1),f(v_2),...,f(v_n)$} is a spanning set ...
4
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3answers
34 views

A direct proof on if $(X, ||\cdot ||)$ is a normed vector space and $Y\subset X$, with $Y$ having finite dimension, then $Y$ is closed.

I am trying to produce a direct proof on the statement mentioned above. The field I am working in is $\mathbb{R}$. My proof outline goes as following: If $Y$ is finite-dimensional, there exists a ...
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3answers
35 views

Is set of all contiuous functions subspace?

This is one of the problems from the book: Hoffman and Kunze, chapter: Vector Spaces Let V be the (real) vector space of all functions f from R into R. Is the set of all f which are continuous, ...
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68 views

Find a linearly independent set of vectors that spans the same subspace of $\mathbb{R}^3$ as that spanned by u, v and w. [on hold]

Consider the vectors $u= (-2, -2, 2)$, $v=(-1, 2, -3)$ and $w=(-6, 0 -2)$. Find a linearly independent set of vectors that spans the same subspace of $\mathbb{R}^3$ as that spanned by $u$, $v$ and ...
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0answers
28 views

space of solutions of a PDE

so I have just completed part (c) and I'm now on part (d). To fill you in, I have found that $K = -\pi^2 (n^2+m^2)$ for some $n,m \in \mathbb{Z}$ and $p = n\pi, q = m\pi$ now I don't really ...
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1answer
33 views

Subspaces and annihilators

I am trying to show this question. My understanding of annihilators is that for a vector space $V$ over $K$, with $S$ being a subset, the annihilator of $S$ is the subspace $S^0$ of linear functions ...
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1answer
42 views

What is the dimension of the vector space of functions $f:\mathbb R\to\mathbb R$?

What is the dimension of the vector space of functions $f:\mathbb R\to\mathbb R$? I want to say that it is at least $2^{\aleph_0}$, but I have no idea how to sharply pin it down otherwise.
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1answer
16 views

Linear functionals and integration verification

Can you please verify my reasoning? (a) Yes as (b) No, as function is squared (c) Yes, same reasoning as (a), squared values of x do not affect linearity. Does the region of integration affect ...
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2answers
56 views

the set of points equidistant from $ u $ and $v$ form a line.

Let $u$ and $v$ be two vectors in $ \mathbb{R}^2 $ with the standard norm. Show that the set of points equidistant from $ u $ and $v$ form a line. I show that if $x$ is equidistant from $u$ and $v$, ...
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0answers
11 views

Where can I find information related to euclidean spaces?

Can you please list some sources where I can study the euclidean space(I am a beginner). Sincerely, I've been trying to understand its meaning and all these symbols, but even the material from the ...
1
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1answer
12 views

Spans containing line through the origin in direction of vector in the set of the span.

span{u,v} contains the line through the origin in the direction of u. TRUE OR FALSE? The solution manual: "True; the span of u is included in the span of u and v." My answer: FALSE. u and v could ...
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0answers
24 views

Proof involving projections and column spaces

Let $A \in \mathbb{M}_{m×n}(\mathbb{R})$ with linearly independent columns. If $\overrightarrow{b} \in \mathbb{R}^m$, then prove $proj_{Col(A)}(\overrightarrow{b}) = ...
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1answer
33 views

Why quotient space is needed?

I was wondering why quotient space is so important? Let say for vector space why quotient space is needed? Please explain!