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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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
46 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 ...
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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!
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1answer
34 views

Any isometry is an isomorphism, though the converse is not true. [closed]

If we define a mapping $f:E \rightarrow F$, where $E$ and $F$ are normed vector spaces, then $f$ is an isometry if $f$ is a linear norm-preserving bijection, that is: $\|f(x)\|=\|x\|, \quad \forall x ...
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1answer
83 views

If $f\in A_k(V)$ and $g\in A_l(V)$ show $i_v(f\wedge g)=i_vf\wedge g+(-1)^kf\wedge i_vg$ - I've got the gist, not sure how to write

If $f\in A_k(V)$ and $g\in A_l(V)$ show $i_v(f\wedge g)=i_vf\wedge g+(-1)^kf\wedge i_vg$ With $A_k(V)$ being the vector space of alternating k-tensors. for $f\in A_k(V)$ for some $v\in V$ we define ...
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3answers
38 views

If I have that $L(v)=L(w)$ for all $L \in V^*$, can I conclude that $v=w$?

Let V be a finite dimensional vector space. If I have that $L(v)=L(w)$ for all $L \in V^*$(where $V^*$ is the dual space of V), can I conclude that $v=w$?
3
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1answer
43 views

Canonical isomorphism between vector bundle and dual?

So, we've been asked to show, given a real vector bundle equipped with a metric, that there is a canonical isomorphism from the vector bundle and its dual. Now, there's a theorem that says two vector ...
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0answers
28 views

A counter example of the direct sum of sub spaces.

I was asked to give examples of 3 subspaces where W + V + U is not the direct sum of these 3 subspaces. W, V, and U are subspaces of a vector space, just to clarify. I am having trouble finding out ...
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2answers
17 views

Example of two subspaces $W_1$ and $W_2$ of $ V$ such that $W_1∪W_2$ is also a subspace of $V$

So I know the obvious counter example would be to let: $W_1 = \{(a, 0) | a \in\mathbb{R}\}$ and $W_2 = (0, 0)$. Where $W_1 + W_2 = (a, 0)$ which is an element of $W_1\cup W_2$. But if I wanted ...
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2answers
28 views

Find a value of $t$ such that the given vector is parallel to $(2,-3,1)$.

Find a value of $t$ such that the vector $(t^2, -3t, (6-t)^{1/2})$ is parallel to $(2,-3,1)$. I'm pretty sure a vector is parallel if it's a scalar multiple of the other, so I tried setting each ...
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0answers
58 views

My answer to this simple question is wrong but I don't know why

I'm self-studying abstract algebra, and prior to fields there's a brief section on vector spaces. One of the questions asks: "Is $U = \{(a, b-1, c)| a, b, c \in F \}$ a subspace of $F^3$? ($F$ a ...
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0answers
20 views

vector space differential equations

Hi! I am working on some differential equations homework and we are up to the linear algebra part. This particular homework set on Vector space is due, but my teacher has not taught the material yet ...
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3answers
51 views

Find a vector in the matching dimension that is not in the span

I have the following vector $(1,2,-2),(2,-1,1)$. How do I find a vector that is not in the span of those two vectors. I can pick an arbitrary third vector and make the other two vectors equal to it ...
2
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1answer
26 views

Finding dimension of a vector subspace/vectorspace

Let $V$ a vector space with dimension $5$ over a field $\mathbb{F}$ and let $W$ a vector subspace of $V$ with dimension $2$. Define $S=\{T:V\rightarrow V \ |\ T \mbox{ is zero on W}\}$ where $T$ is a ...
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0answers
17 views

Question on the space of all square summable functions involving operator norms and eigenvalues.

Recall that for a set M, $ \mathscr ℓ^2 (M) $ is the space of all square summable functions M $ \to \Bbb C $ . Let $\mathrm T \in Hom( ℓ^2 (\Bbb N) , ℓ^2 (\Bbb N)) $ be given by $$ (\mathrm T a)(n) ...
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0answers
18 views

Proving that $\left(X(v,v)\right)^{\frac{1}{2}}$ defines a norm on a vector space where X is a positive-definite bilinear form.

Want to show: If $X$ is a positive-definite bilinear form on a vector space $G$ with real-valued scalars and $v\in G$, then $\left(X(v,v)\right)^{\frac{1}{2}}$ defines a norm on $G$. Thus far I have ...
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1answer
16 views

Proving the dimension of the intersection of 2 subspaces

Assume that $U$ and $W$ are distinct subspaces $( U ≠ W )$ of a four-dimensional vector space $V$ and $\dim(U) = \dim(W) = 3$. Prove that $\dim ( U ∩ W ) = 2$.
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4answers
37 views

Proof these operations is not a vector space

Let $\mathbb{R}$ be the set of all real numbers. Define scalar multiplication by $\alpha x = \alpha \cdot x$ (the usual multiplication of real numbers) and define addition by $x \oplus y = \max(x, ...
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2answers
16 views

Notation for the “scalarization” of a vector with a single non-zero entry

Suppose I have a vector $v$ in the complex space $\mathbb{C}^N$ with only a single non-zero element. Is there a standard notation to replace the vector with a scalar equal to the non-zero value of ...
1
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2answers
51 views

Proof that $[v, Tv, T²v, … , T^n v]$ is a basis for $V$ ($dim(V)=n$)?

Let $T:V\to V$ be a linear map from a finite dimensional vector space over a field $F$ to itself. Assume $[v,Tv,T²v,...]$ spans $V$ for some $v \in V$. Don't know at all how to prove that $[v, Tv, ...
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1answer
33 views

Prove that $z = tx + (1 − t)y$ if $d(x, y)= d(x, z) + d(z, y)$

Let $x,y,z$ be elements of $\mathbb{R}^2$ Prove that $z = tx + (1 − t)y$ if $d(x, y)= d(x, z) + d(z, y)$ d is usual euclidean metric.
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3answers
44 views

If T is diagonalizable then prove that T inverse is diagonalizable. [closed]

If T is an invertible linear operator on a finite dimensional vector space V, then if T is diagonalizable prove that T inverse is also diagonalizable.
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2answers
29 views

How: $ (\vec b + \vec c)(\vec b - \vec c) = 0 \implies \frac{\vec b + \vec c}{2}.(\vec c - \vec b) = 0 $

I am having difficulty understanding one step of the solution to a question. question:- Using vectors, prove that the median to the base of an isosceles triangle is perpendicular to the base. ...
3
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1answer
48 views

Prove { T₁, . . . , Tᵣ } is a basis for V*

Let V be a vector space over a field F, and let α = { v₁, . . . , vᵣ } be a basis for V . For each 1 ≤ i ≤ r, define Tᵢ: V → F by Ti(a₁v₁ + · · · + aᵣvᵣ) = aᵢ Prove that { T₁, . . . , Tᵣ } is a basis ...
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0answers
37 views

Linear Alegbra - The $U + W$, $U \cap W$, $U \cup W$, $U-W$ of subspaces and not subspaces

Lets assume $U,W \subseteq \mathbb{R}^4$ $U=\{u_1 = (0,0,0,1), u_2=(1,0,0,0)\}$ $W=\{w_1=(0,0,1,0),w_2=(1,0,0,0),w_3=(0,1,0,0)\}$ I understand that in case $U$ and $W$ are not subspaces: Case $U ...
3
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2answers
34 views

Prove a linear transformation symmetric and positive

Consider the linear transformation of $\mathbb{R}^3$ given by $Ax = (a \cdot x)a+ |a|^2x$. Is $A$ symmetric? Is it positive? I know that a matrix is symmetric IFF $(Ax,y) = (x,Ay)$ and positive if ...
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1answer
21 views

Is this set a subspace?

Let $\mathbb{F}[X]$ be the vector space of all polynomials over the field $\mathbb{F}.$ I'm trying to determine whether the following set is a subspace: $$W=\{p:p(0)=p(1)\}$$ where $p$ is a ...
3
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0answers
48 views

Find the projection of any vector onto the linear span and the normal from any vector to that span

Show that the vectors $u_1 = (1/9,4/9,8/9), u_2=(8/9,-4/9,1/9), u_3=(-4/9,-7/9,4/9)$ form an orthonormal basis of $\mathbb{R}^3$. Find the projection of any vector $x=(\xi_1,\xi_2,\xi_3) \in ...
0
votes
1answer
25 views

How many vectors exist satisfying the angle between any two vectors equals to a constant $\beta$ with $0<\beta<\pi$ in a $n$-dimension Euclid space?

At first, if $\beta=\pi/2$, we know that at most $n$ such vectors exist, that is, orthogonal vectors. It's obvious that the number of vectors is influenced by the angle $\beta$. Assume we've already ...
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0answers
15 views

Need help in understanding the vector space answer for the last part

The linear transformation $T : \mathbb{R}_4 \to\mathbb{R}_4$ is represented by the matrix $$A = \begin{bmatrix}1 &&−1 &&2 &&3\\ 2&& −3 &&4 &&5\\ 5 ...
3
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1answer
21 views

How to find dimension of a subspace?

For instance, take $S=\{\mathbf{v}\in\mathbb{R}^5:\;v_1+v_2+v_3=0,\;v_1+v_2+v_5=0\}\subset\mathbb{R}^5.$ How would I go about finding $\dim S?$ I can see that both $v_1+v_2+v_3=0,\;v_1+v_2+v_5=0$ ...
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1answer
32 views

Unknown Charge on Rectangle?

A charge is to be placed at the empty corner to make the net force at corner A point along the vertical direction. What charge (magnitude and algebraic sign) must be placed at the empty corner if the ...
2
votes
2answers
128 views

Vector space as simple $K[x]$-module

I am trying to solve the problem: Let $V$ be a vector space and $T$ a linear transformation $T:V \to V$. Let $(V,T)$ be a $K[x]$-module. Show that $(V,T)$ is simple if and only if $V$ is finite ...
3
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1answer
37 views

Existence of non-trivial linear functional on any vector space

For every vector space $V$ does there exist a linear functional $f$ ( a linear map from $V$ to $F$ the underlying field ) such that for some $ \vec v \in V$ , $f(\vec v) \ne 0$ ? If it does exist , ...
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2answers
31 views

Find quotient of vector spaces [closed]

How can I find the solution of this quotient $$\frac{Span\{a,b,c,d\}}{Span\{a+b,c+d\}}$$?
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2answers
27 views

Proof that $(-\lambda)v=-(\lambda v)$ and $\lambda(-v)=-(\lambda v)$

Let $V$ be a vector space over a field $\mathbb{F}$, and let $\lambda \in \mathbb{F}$ and $v \in V.$ I'm trying to prove the following results (using the vector space axioms): 1) ...
0
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2answers
41 views

Is it true that $\langle a+cb, a+cb\rangle = \langle a,a\rangle + c^2 \langle b,b\rangle $ where $a,b \in \mathbb{R}^n$ and $c \in \mathbb{R}$?

We have $a=(a_1, \dots \dots , a_n)$ and $b=(b_1, \dots \dots , b_n) \quad$ so $a+cb = (a_1+cb_1, \dots \dots , a_n+cb_n) \quad$ and $\langle a+cb, a+cb\rangle = (a_1+ cb_1)^2 + \dots \dots + ...
0
votes
1answer
38 views

Linear Algebra Dimension

Let $L(U,V)$ = $\{T:U\rightarrow V\ :\ T\ \text{linear}\},$ and dim $(U)=n$, dim $(V)=m$. Then show that $$ \dim L(U,V) = mn. $$ I don't know how to begin and I already searched the internet to find ...
1
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1answer
23 views

Quotient ring of a graded algebra with respect to a graded ideal

An algebra $A$ over $F$ is said to be a graded algebra if as a vector space over $F$, $A$ can be written in the form $$A=\bigoplus_{i=0}^\infty A_i$$ for subspaces $A_i$ of $A$ along with other ...
3
votes
3answers
267 views

What is the main difference between a vector space and a field?

In my opinion both are almost same. However there should be some differenes like any two elements can be multiplied in a field but it is not allowed in vector space as only scalar multiplication is ...
0
votes
1answer
18 views

Determining if a point is on which side of a line (explanation)

Source: http://datasciencelab.wordpress.com/2014/01/10/machine-learning-classics-the-perceptron/ I asked this question on stack overflow, but I guess it is more appropriate here. "The general ...
1
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1answer
47 views

Distinction between point and vector outside of US ( particularly Germany and Eastern Europe )

There was a long discussion in a forum I visit in where a calculus teacher was being critical of Stewarts Calculous for making a distinction between points and vectors. He argued that no such ...
2
votes
3answers
47 views

What is the dimension for this subspace?

For V (x,y,z) $S$ is a subspace of $V$ satisfying following condition. $x+y+z=0$ $x+y+z=0$ and $x-y-z=0$ I dont know about 1 but For number 2, isn't $x$,$y$ and $z$ $0$? then dimension of ...
1
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0answers
20 views

Proving that a quotient space is formed from a vector space with W-affine subspaces.

I have been given a 2-part question which first states given a vector space (V,K) and W$\subseteq$V is a subspace. that a W-affine subspace S$\subseteq$V is one in which s,s' $\in$ S, s-s' $\in$ W and ...