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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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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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 ...
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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 ...
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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 ...
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1answer
26 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 ...
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Show that $P_n$ is an $(n+1)$-dimensional subspace of the vector space of all real polynomials

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 ...
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24 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 ...
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0answers
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Is it always possible to find the Reduced Row Echelon form of a matrix, given the basis of its null space?

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.
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$X$ 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 ...
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$m_p=\{f\in \mathcal{O}_{V,p}| f(p)=0\}$, 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 ...
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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 ...
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1answer
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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 $$ < ...
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1answer
50 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 ...
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1answer
15 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 ...
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2answers
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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 ...
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1answer
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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, ...
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1answer
29 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, ...
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28 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 ...
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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 ...
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1answer
14 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 ...
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2answers
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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 ...
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1answer
34 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?
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1answer
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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 $$(a_2x^2+a_1x+a_0)+(b_2x^2+b_1x+b_0)=(a_2+b_2)x^2+(a_1+b_1)x+(a_0+b_0)$$ ...
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1answer
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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 ...
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1answer
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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, ..., ...
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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}\}$ ...
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1answer
16 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 ...
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0answers
8 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?
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2answers
38 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 ...
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1answer
33 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 ...
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3answers
37 views

The union of three subspaces equals to a vector space

I am aware that the union of subspaces does not necessarily yield a subspace. However, I am confused about the following question: (i) Let $U, U'$ be subspaces of a vector space $V$ (both not ...
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3answers
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Proof: dimension of the vector space of solutions to the system Bx=0

I've run into a matrix dimension proof I'm having some trouble with: Let $A,B$ be $n\times n$ matrices, and let $P(A),P(B),P(AB)$ be the vector spaces of solutions to the systems ...
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Finding ranks and nullities of linear maps

I am confused about ranks, nullities and bases of the kernel. From what I understand the rank is the dimension of a vector space generated by a matrix. How would I do the following examples? Find ...
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2answers
22 views

Describe all vectors $v = \pmatrix{x\\y}$ that are orthogonal to $u = \pmatrix{a\\b}

Describe all vectors $v = \pmatrix{x\\y}$ that are orthogonal to $u = \pmatrix{a\\b}$. I know that vectors that are orthogonal will have a dot product of 0. So here's what I was thinking: ...
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2answers
27 views

Finding a linear transformation with a given null space

The problem statement is, Find a linear transformation $T: \mathbb R^3 \to \mathbb R^3$ such that the set of all vectors satisfying $4x_1-3x_2+x_3=0$ is the (i) null space of $T$ (ii) range of $T$. ...
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1answer
27 views

Relating regression to projection?

I recently learned that one can think of regression as a projection of a vector in a high dimension space onto the other vector. I tried implementing this and got it to work: ...
2
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1answer
33 views

When is a manifold also a vector space?

My question arises from this definition: Poincare group is the group of Minkowski space-time isometries. Which means that it leaves the space-time intervals unchanged. Now here is my understanding: ...
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1answer
26 views

Dimension of the subspace of a vector space spanned by the following vectors.

I know that in order to find a subsequence that is a basis of a subspace is to check whether the given vectors are linearly independent and whether they span the subspace. However how can I find the ...
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1answer
26 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 ...
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1answer
14 views

Write $F$ as a linear combination of elements of $\mathcal B^*$

If $V=\mathbb R[x]_k=\{\sum\limits_{i=1}^ka_ix^i:a_i\in\mathbb R, \forall i\}$ is a vector space of dimension $k+1$ over $K=\mathbb R$ and $\mathcal B=\{1,x,\dots,x^k\}$ is a basis of $V$. The dual ...
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42 views

Describing all the linear transformations satisfying the constraints

How to find the linear transformation $T: \mathbb R^3 \to \mathbb R^3$ such that the set of all vectors satistfying $4x_1-3x_2+x_3=0$ is a) Null space of $T$ b) Range of $T$ I'm not able to ...
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1answer
23 views

Proving a basis spans $R^3$

Doing some reviewing and I'm not 100% sure if my thought-process is correct. I have the following two vectors and need to prove they're a basis for $R^3$: $$B= \begin{bmatrix} 1 \\ ...
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0answers
17 views

satellites attitude determination TRIAD - how are orbital reference frame vectors constructed?

I posted this same question on space.stackexchange but never received any answer. So I am posting here hoping to get an answer as this is a quite mathematical topic. I am trying to fully understand ...
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1answer
11 views

Function Inequality

Let $E$ and $F$ be normed vector spaces and $\mathscr{L}(E,F) = \{f:E \rightarrow F \mid f$ is linear and continuous$\}$ be a normed vector space with the norm $\lVert f \rVert = \sup_{|x|=1} \{|f(x)| ...
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Find the dimensions spanned by the vectors [on hold]

Compute the dimension of the subspace spanned by each subset. $\{1, e^{ax}, xe^{ax}\}, \{1, \cos 2x,\sin^2 x\}$
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0answers
21 views

Defining a function on a Euclidean Vector Space

Let $\{x, y\}$ be a linearly independent set in a Euclidean space V. Define $f : \mathbb{R} \rightarrow \mathbb{R}$ by $f(a) = \|x − ay\|$ I do not understand the question. How should I define that ...
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1answer
28 views

The set $W^{⊥⊥}$ in a Hermitian space

Problem Statement: Let $W$ be a subspace of a Hermitian space $V$. Prove that $W^{⊥⊥}=W$ I am trying to figure out a good strategy for this proof. I know that: $W$ is a subspace of $V$ ...
0
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1answer
14 views

problem on annihilators on finite dimensional spaces

Suppose $V$ and $W$ are subspaces of a finite-dimensional vector space $U$. Show that if $V^0 \subset W^0$ then $W \subset V$ This is an exercise problem in Linear Algebra Done Right, 3rd ...
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1answer
30 views

“vector” vs “point” in definition of directional derivative

Given a function $f\colon \mathbb R^n\to\mathbb R$, and given $x,v\in\mathbb R^n$, it is customary to define the "directional derivative of $f$ in the direction $v$ at the point $x$" by $$ D_v f(x) = ...
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2answers
22 views

Direct Sum of $n$ Subspaces

I just need some guidance to prove a portion of the following theorem. Let $V_1, V_2, ... , V_n$ be subspaces of a vector space $V$. Then the following statements are equivalent. $W = \sum ...