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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22 views

The sum of two subspaces

Let $V_{1}$ and $V_{2}$ be two subspaces of V. Define the sum of $V_{1}$ and $V_{2}$ to be the subset of V $V_{1}+V_{2}=${$\overrightarrow v_{1} + \overrightarrow v_{2}:\overrightarrow v_{1} \in ...
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0answers
7 views

How can I find the stability of the equilibria of this vector field?

Consider the vector field given by $y' = y - y^{3}$. This clearly has equilibria at the points $y = 0, \; y = 1, \; y = -1$. How would I find the stability of these points though? I understand that I ...
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1answer
20 views

Basis of vector space. [on hold]

If $B$ is a subset of a vector space $V$ such that $B$ spans $V$ but no proper subset of $B$ spans $V$ then prove that $B$ is a basis of $V$.
2
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1answer
22 views

A linearly independent set that spans a space

So, in partial differential equations, we generate solutions for PDEs (kind of obviously). However, while the solutions we generate span the space of all solutions and are all linearly independent, ...
0
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3answers
50 views

Linear maps (about rank and nullity)

$S:U\rightarrow V$ and $T:V\rightarrow W$ are linear maps. $U,V$ and $W$ are vector space over the same field. Prove: If $V=W$ and $T$ is non-singular then ...
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0answers
11 views

Equal hyperplanes implies proportional coefficients

Let $E$ be a $K$-vector space with basis $(e_1,\cdots,e_n)$, such that for any $x=x_1e_1+...+x_ne_n$. Let $a=(a_1,\cdots,a_n)\in K^n$ and $b=(b_1,\cdots,b_n)\in K^n$. Define the hyperplanes ...
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1answer
35 views

Water main construction. Find the angle using vectors.

A water main is to be constructed with a $12.5$​% grade in the north direction and a $25$​% grade in the east direction. Determine the angle $\theta$ required in the water main for the turn from north ...
3
votes
1answer
33 views

Solve the following vector equations simultaneously $\vec x+\vec c \times \vec y=\vec a $ and $\vec y+\vec c \times \vec x=\vec b $.

Solve the following vector equations simultaneously $\vec x+\vec c \times \vec y=\vec a $ and $\vec y+\vec c \times \vec x=\vec b $. I tried $$\vec c \times (\vec x+\vec c \times \vec y)=\vec c ...
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1answer
23 views

Isomorphism is an equivalence relation on finite dimensional vector spaces over $F$.

Show that isomorphism is an equivalence relation on finite dimensional vector spaces over $F$. A relation $R$ is an equivalence relation if it is: reflexive, i.e. $xRx$ for all $x$ symmetric, ...
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1answer
21 views

A Mapping from a Power Sets of a Vector Space to a Set of Subspaces of a Vector Space

I don't necessarily have a question on how to approach the problem. In this post, I want to get some clarification on how the problem is defining a certain function. The following question was given ...
0
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1answer
24 views

Elements fail to form a basis

Consider the vector space $P$2 and the set $$5−1t+4t^2,−4+3t+1t^2,8+5t+kt^2$$ For which $k \in \mathbb{R}$, do these three elements fail to be a basis of $P$2? I thought in order to make the three ...
1
vote
1answer
36 views

Why does the additive inverse not follow

I need to prove that the vector space of $\mathbb{R}^2$ with the following operations: $x + y = (x_1 + 2y_1, 3x_2 - y_2)$ The usual scalar multiplication of $cx = (cx_1, cx_2)$ The answers in my ...
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vote
2answers
44 views

Expressing linear transforms using linear functionals: is this possible?

Work over a fixed but arbitrary field. Let $Y$ and $X$ denote finite-dimensional vectorspaces, and let $y \in Y^n$ denote a sequence of elements of $Y$, where $n$ is a natural number. It seems ...
0
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1answer
48 views

Why is the statement “all vector space have a basis” is equivalent to the axiom of choice? [duplicate]

I'm reading a section in an abstract algebra book, where it reviews vector spaces and suddenly comments that "all vector space have a basis" is equivalent to the axiom of choice...I haven't studied ...
0
votes
1answer
17 views

Show that for each $v \in V$ exist $w \in W$ and $c \in \Bbb R$ unique such that $v=cv_0+v$

Let be $V$ a vector space over the field of real numbers, $f \in V^*$ and $W=ker (f)$. If $v_0 \in V$ is a vector such that $f(v_0)\neq0$, show that for each $v \in V$ exist $w \in W$ and $c \in \Bbb ...
2
votes
1answer
24 views

Finite abelian group as Z-module

If $M$ is a finite abelian group then $M$ is naturally a $Z$-module. Can this action be extended to make $M$ into a $Q$-module ?
1
vote
1answer
17 views

''Linear'' transformations between vector spaces over different fields .

Let $\mathbf{V}(\mathbb{K}_1,V),$ and $\mathbf{W}(\mathbb{K}_2,W)$ two vector spaces over different fields ( as an example: $\mathbb{K}_1=\mathbb{C}$ and $\mathbb{K}_2=\mathbb{R}$). We can generalize ...
0
votes
1answer
29 views

Is $x^4 + y^2 + z^6 = 0$ a subspace of $R^3$?

I have already verified it contains the zero vector, and gone through the standard showing that $\alpha x_1 + \beta x_2 \in S$, however I can't help but think there's a counterexample, because surely ...
2
votes
0answers
20 views

dimension and base of arithmetic sequence

Arithmetic sequence is a vector space. But how to find a dimension of it. I try this: arithmetic sequence: $a_1,a_2,...,a_n={a_1,(a_1+d),(a_1+2d),...,(a_n+(n-1)d)}$ With only $a_1$ and $d=(a_1-a_2$) ...
0
votes
1answer
21 views

Drawing half a circle betweeen two arbitrary 2D points

So, I have two arbitrary points in a vector space and I'm trying to draw 180 degrees of a circle between them. The radius of the circle would be half of the distance between the two points and the ...
0
votes
1answer
19 views

eigenvalue and rank of a transformation

what i feel is that since the range of the linear transformation is strictly less than $n$ this implies that the transformation is not onto hence the null space contains a non trivial vector.but is ...
0
votes
1answer
68 views

Define $\phi:\mathbb{R}^3 \rightarrow \mathbb{R}$ by $\phi(e_1) = 1$, $\phi(e_2) = 2$, $\phi(e_3)=-1$. Determine ker$\phi$ and im$\phi$

Let {$e_1,e_2,e_3$} be the standard basis for $\mathbb{R}^3$ and define $\phi:\mathbb{R}^3 \rightarrow \mathbb{R}$ by $\phi(e_1) = 1$, $\phi(e_2) = 2$, $\phi(e_3)=-1$. Determine the subspaces ...
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0answers
12 views

Linear Transformations from $\mathbb{P}_n(t)$ to $\mathbb{P}_n(t)$

I want to check to see if I am understanding Linear Transformations. I have the following problem that was given to us in lecture to do for practice. For a vector space $\mathbb{P}_n(t)$ over ...
0
votes
1answer
29 views

Show that $Im T$ and $U/Nuc T$ are isomorphic for a linear transformation $T: U \longrightarrow V$

Show that $Im$ $T$ and $U/Nuc$ $T$ are isomorphic for a linear transformation $T: U \longrightarrow V$ Hi guys, I know how to show this for vectorial spaces with finite dimension, but I don't have ...
0
votes
2answers
34 views

How to find $\dim W_1$, $\dim W_2$, $\dim W_1+W_2$, $\dim W_1\cap W_2$ for the following spans?

Let $W_1=\{(1,1,2,1), (3,1,0,0)\}$ and $W_2=\{(-1,-2,0,1), (-4,-2,-2,-1)\}$ Apparently $\dim W_1=\dim W_2=2$. For $\dim W_1\cap W_2$, since $(-4,-2,-2,-1)$ can be expressed as ...
0
votes
1answer
56 views

$\mathbb C$-dimension of vector space $\mathbb C\otimes_{\mathbb R}\mathbb C$

Let $\mathbb R$ be the field of real numbers, $\mathbb C$ be the field of complex numbers. Consider $\mathbb C\otimes_{\mathbb R}\mathbb C$ as a $\mathbb C$-vector space via $a(b\otimes c) := ab ...
0
votes
1answer
63 views

Show that $P_n$ is an $(n+1)$-dimensional subspace of the vector space of all real polynomials [duplicate]

Show that $P_n$ = {polynomials with real coefficients of degree $\leq$ n} is an ($n+1$)-dimensional subspace of the infinite-dimensional vector space of all real polynomials I know that $P_n$ is ...
0
votes
1answer
29 views

Dimension of the image of a matrix

So the question asks: Verify if the image of the linear map $T : \mathbb{R}^6 \to \mathbb{R}^3$ given by left multiplication by A= $$\begin{bmatrix}6 & 0 &2 & 2& 3& 4\\0 & -1 ...
0
votes
0answers
13 views

Is it always possible to find the Reduced Row Echelon form of a matrix, given the basis of its null space? [on hold]

I tried starting with multiple bases of the null space and each time I was able to write the RREF form of the matrix. However, I have not been able to prove that this is true for all possible bases.
0
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1answer
16 views

$X$ is a normed linear space such that for some compact $K\subseteq X$ , $\operatorname{span} K$ is dense in $X$ then is $X$ separable?

Let $X$ be a normed linear space which is separable. Then I know that there exists a compact subset $K$ of $X$ such that $\operatorname{span} K$ is dense in $X$ (in fact we can also find compact and ...
4
votes
1answer
46 views

$m_p=\{f\in \mathcal{O}_{V,p}| f(p)=0\}$, ideal of $p$ in the local ring. What is $m_p/m_p^2$?

In Section 6.8 of Undergraduate Algebraic Geometry by Reid, the author proved the following Theorem: There is a natural isomorphism of vector spaces $(T_pV)^*\cong m_p/m_p^2$ where $^*$ denotes ...
0
votes
1answer
18 views

How to show map is non-singular

Let $f:\;\mathbb{R}^n\to\mathbb{R}^n$ be differentiable. Suppose that for all $x\in\mathbb{R}^n:$ $$\lVert \mathrm{D}f(x)-\mathrm{I}\rVert\leq \frac{1}{2}$$ where $\lVert\cdot\rVert$ is the ...
1
vote
1answer
22 views

Solving Laguerre coefficients with Integral?

I'm having some difficulty understanding the solution to a particular Laguerre expansion. The problem reads "Expand the term $ e^{-x}$ as a Laguerre expansion, noting the orthogonality of $$ < ...
2
votes
1answer
54 views

group action same thing as homomorphism

A linear group action of a group $G$ on a vector space $V$ is the same thing as a homomorphism from G to the general linear group $GL(V)$. attempt: Suppose a linear group action of a group $G$ on a ...
1
vote
1answer
16 views

How to show $S_1\subset W_1$ and $S_2\subset W_2$ are independent $\implies$ $S_1\cup S_2$ is independent based on the following assumption?

Let $W_1$ and $W_2$ be subspaces of vector space $V$ satisfying $W_1\cap W_2=\{0\}$ ,how to show $S_1\subset W_1$ and $S_2\subset W_2$ are linearly independent $\implies$ $S_1\cup S_2$ is linearly ...
1
vote
2answers
17 views

Is $S_1\cap S_2$ and $S_1\setminus S_2$ always linearly dependent if $S_1$ and $S_2$ are linearly dependent subsets of vector space $V$?

Let $S_1$ and $S_2$ be linearly dependent subsets of vector space $V$, are $S_1\cap S_2$ and $S_1\setminus S_2$ always linearly dependent? The counterexample for the first one I can think of is ...
0
votes
1answer
21 views

How to show $P_3(R)=W\oplus W_1$ and $P_3(R)=W\oplus W_2$ based on the following assumption?

Let $W=$Span$\{1, x\}$, $W_1=$Span$\{x^2, x^3\}$ and $W_2=$Span$\{1+x+x^2+x^3, 1+x+x^2-x^3\}$, how to show $P_3(R)=W\oplus W_1$ and $P_3(R)=W\oplus W_2$? $P_3(R)=W+ W_1$ because Span$\{1, ...
0
votes
1answer
31 views

How to find $U+W$ and $U\cap W$ based on the following assumption?

Let $U=\{(x_1, x_2, x_3, x_4)\in R^4\mid x_1+ x_2=0, x_3+ x_4=0 \}$, $W=\{(x_1, x_2, x_3, x_4)\in R^4\mid x_1+ x_3=0, x_2+ x_4=0 \}$, how to find $U+W$ and $U\cap W$? I think $U\cap W=\{(x_1, x_2, ...
0
votes
1answer
29 views

Find the dimension and a basis of a subspace

Let $U$ is the set of all commuting matrices with matrix $A= \begin{bmatrix} 2 & 0 & 1 \\ 0 & 1 & 1 \\ 3 & 0 & 4 \\ \end{bmatrix}$. Prove ...
0
votes
0answers
15 views

How to show that $V=$Span$(S_2)$ if Span$(S_1)=V$ and that every vector in $S_1$ is in Span$(S_2)$?

Let $S_1$ and $S_2$ be subsets of a vector space $V$. Assume Span$(S_1)=V$ and that every vector in $S_1$ is in Span$(S_2)$, how to show that $V=$Span$(S_2)$ as well? In my opinion, to show ...
1
vote
1answer
16 views

How to show that $R^3$ is the direct sum of $W_1=$Span$(1,1,1)$ and $W_2=$Span$(\{1,0,0\}, \{1,1,0\})$?

How to show that $R^3$ is the direct sum of $W_1=$Span$(1,1,1)$ and $W_2=$Span$(\{1,0,0\}, \{1,1,0\})$? So we write it as $R^3=W_1+W_2$ because every $(x_1, x_2, x_3)\in R^3$ can be written as ...
0
votes
2answers
16 views

How to prove that $W_1\cap W_2\supset$ Span$(S_1\cap S_2)$ if $W_1=$ Span$(S_1)$ and $W_2=$ Span$(S_2)$ are subspaces of vector space?

In my opinion, let $v\in$ Span($S_1\cap S_2$) and therefore $v\in$ Span$(S_1)$ and $v\in$ Span$(S_2)$. Write $v=c_1z_1+...+ c_nz_n$ where $z_k\in S_1\cap S_2$ and $c_k\in R$. Here I am feeling I have ...
1
vote
1answer
36 views

If commutativity of vector space is omitted, can we still use other axioms to prove the commutativity?

Here I am thinking of using $-(x+y)$ and show that it equals $-(y+x)$. $-(x+y)=-x-y$ by distributivity =$-x-y+0=...$ Here I don't know how to continue, could someone suggest?
0
votes
1answer
20 views

How to show that the following is satisfied for all vector space axiom?

Let $V=\{a_2x^2+a_1x+a_0|a_1, a_2, a_3\in \mathbb{R}, a_2\ne 0\}$ with operation defined by $$(a_2x^2+a_1x+a_0)+(b_2x^2+b_1x+b_0)=(a_2+b_2)x^2+(a_1+b_1)x+(a_0+b_0)$$ ...
0
votes
1answer
27 views

How to show that $M_{2\times 2}(\mathbb{R})=W_1\oplus W_2$ based on the following assumption?

Let the subspaces $W_1=\{\begin{pmatrix}a&b\\-b&a \end{pmatrix}|a, b\in \mathbb{R}\}$ and $W_2=\{\begin{pmatrix}c&d\\d&-c \end{pmatrix}|c, d\in \mathbb{R}\}$ of $M_{2\times ...
1
vote
1answer
17 views

If $x+y=(x_1y_1, …, x_ny_n)$ and $c\cdot '\ x=x^c_1, …, x^c_n$, how to show that with these two operation $V$ is a subspace?

Let $V=(R^+)^n=\{(x_1, ..., x_n)| x_i\in R^+$for each $i\}$. In $V$ define a vector sum operation $+'$ by $x+y=(x_1y_1, ..., x_ny_n)$ and scalar multiplication $\cdot '$ by $c\cdot '\ x=x^c_1, ..., ...
1
vote
2answers
24 views

How to show if the following subset $W$ is a subspace of a vector space $V$?

$1.$ $V=P_n(\mathbb{R}), $and $ W=\{p(x)\in P_n(\mathbb{R})\mid p(1)+p(2)+p(3)=0 \}$ $2.$ $V=M_{n\times n}(\mathbb{R}), $and $ W=\{A\in M_{n\times n}(\mathbb{R}) \mid A \text{ is not symmetric}\}$ ...
0
votes
1answer
17 views

Given a basis $U$, what conditions are needed for an orthogonal basis for it?

Given a basis $U$, what conditions are needed for an orthogonal basis for it? For example, in the following vector space $U$, if $U =sp\{(1,1,1),(1,3,7)\}$ then what conditions are needed for an ...
0
votes
0answers
10 views

how to write amatlab code for document representation using second order tensor [on hold]

My project is on document representation using tensor.How i will represent a document using second order tensor in matlab?
0
votes
2answers
40 views

Linear transformation from a vector space to a field

Can anyone help me with the following question: Let $V$ be a vector space over field $F$, possibly not finite dimensional. Let $T \colon V \to F$ be a linear map. Prove that there is no subspace ...