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 ...

learn more… | top users | synonyms

3
votes
0answers
14 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 ...
0
votes
1answer
19 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, ...
0
votes
1answer
18 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 ...
2
votes
1answer
248 views

Positivity of the alternating sum associated to at most five subspaces

Let $V_1 , V_2 , \dots , V_n $ be proper subspaces of $ \mathbb{C}^m$ and let $$\alpha = \sum_{r=1}^n (-1)^{r+1} \sum_{ \ i_1 < i_2 < \cdots < i_r } dim(V_{i_1} \cap \cdots \cap V_{i_r})$$ ...
0
votes
1answer
17 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
30 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 ...
1
vote
1answer
24 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
votes
1answer
44 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
0answers
29 views

Isolated zero $z$ of $X$ on a star shaped polygon.

Consider a polygon that is star shaped with respect to the isolated zero $z$ of $X$. I want to show that the boundary of the polygon can be made transverse to $X$ by jiggling vertices only in the ...
0
votes
1answer
14 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 ...
1
vote
3answers
245 views

If $M, N$ are finite dimensional vector spaces with same dimension, then if $M$ is subset of $N$, then $M=N$.

If $M, N$ are finite dimensional vector spaces with same dimension, then if $M$ is subset of $N$, then $M=N$. I think i need to show that both vector spaces are spanned by the same bases in order to ...
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
30 views

Find $U\cap V$.

Find $U\cap V$. Given $$U = \text{span}{(1,1,-1),(2,3,-1),(3,1,-5)}$$ $$ V=\text{span} {(1,1,-3),(3,-2,-8),(2,1,-3)}$$ $A. U$ $B. V$ $C. \{0\}$ $D.$ None of the above ATTEMPT: I have found that ...
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
28 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 ...
8
votes
7answers
9k views

Real life applications of general vector spaces

Students familiar with Euclidean space find the introduction of general vectors spaces pretty boring and abstract particularly when describing vector spaces such as set of polynomials or set of ...
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
19 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
17 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
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
58 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
23 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 ...
3
votes
1answer
379 views

Vector Project onto Subspace

So the question is: Let S be the subspace of $\mathbb{R}^3$ spanned by the vectors $ u_2 = \begin{pmatrix} \frac{2}{3}\\\frac{2}{3}\\\frac{1}{3}\end{pmatrix} u_3 = \begin{pmatrix} ...
0
votes
1answer
57 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 ...
0
votes
1answer
52 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 ...
1
vote
0answers
11 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

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
votes
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 ...
2
votes
1answer
53 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 ...
4
votes
1answer
45 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
17 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 ...
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, ...
2
votes
1answer
524 views

Triangle and parametric coordinates

I'm studying on a book where it says: "A triangle is the set of points where for some point po, where u and v range over the parametric coordinates (we are talking about barycentric coordinates ...
1
vote
1answer
21 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 $$ < ...
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
1answer
34 views

Showing that the following vectors are linearly independent in a subspace which they do not span.

I am trying to better understand vector spaces and dimensions. I could prove (i) via induction and the definition of linear independence? However how can I approach the questions (ii),(iii) which ...
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
30 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, ...
1
vote
1answer
15 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, ..., ...
0
votes
2answers
13 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 ...
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 ...
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
19 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
26 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 ...
0
votes
1answer
27 views

Independence, Inverse and Additive Identity for vector with defined vector addition and scalar multiplication

I am struggling to find my bearings on this question. I am confident that I can do parts a and d. I have no clue how to approach b. I was also wondering if the redefined vector addition and scalar ...
2
votes
3answers
4k views

Curl of Cross Product of Two Vectors

I want to prove the following identity $$\text{curl } \left(\textbf{F}\times \textbf{G}\right) = \textbf{F}\text{ div}\textbf{ G}- \textbf{G}\text{ div}\textbf{ F}+ \left(\textbf{G}\cdot \nabla ...
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}\}$ ...