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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Embedding torsion-free abelian groups into $\mathbb Q^n$?

Glass' Partially Ordered Groups states without proof: Every torsion-free abelian group can be embedded into a rational vector space (as a group). Can someone link me to a proof of this? It ...
4
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3answers
96 views

For subspaces, if $N\subseteq M_1\cup\cdots\cup M_k$, then $N\subseteq M_i$ for some $i$?

I have a vector space $V$ over a field of characteristic $0$. If $M_1,\dots,M_k$ are proper subspaces of $V$, and $N$ is a subspace of $V$ such that $N\subseteq M_1\cup\cdots\cup M_k$, how can you ...
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2answers
186 views

The isomorphisms between two vector spaces

Let $V$ and $W$ be two vector spaces over real number field, if they are isomorphic as vector spaces over rational number field, are they isomorphic as real vector spaces ?
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2k views

Span of an empty set is the zero vector

I am reading Nering's book on Linear Algebra and in the section on vector spaces he makes the comment, "We also agree that the empty set spans the set consisting of the zero vector alone". Is Nering ...
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1answer
155 views

Is $\mathbb{R}^1$ a subspace of $\mathbb{R}^2?$

My intuition tells me it is. But in terms of vectors, the span of a vector with only one component (a vector in $\mathbb{R}^1$) is not said to be a subspace of $\mathbb{R}^2$
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2answers
407 views

If a field $F$ is such that $\left|F\right|>n-1$ why is $V$ a vector space over $F$ not equal to the union of proper subspaces of $V$

If $U_1$, $U_2,\ldots,U_n$ are proper subspaces of a vector space $V$ over a field $F$, and $|F|\gt n-1$, why is $V$ not equal to the union of the subspaces $U_1$, $U_2,\ldots,U_n$?
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461 views

Help a newbie understand Linear Algebraic terms

I am taking a class in Algebra but I am having a problem grasping exactly what it is I am being asked to do -- I think I am having a problem with the vocabulary being used. I have a couple of ...
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2answers
122 views

Is an infinite linearly independent subset of $\Bbb R$ dense?

Suppose $(a_n)$ is a real sequence and $A:=\{a_n \mid n\in \Bbb N \}$ has an infinite linearly independent subset (with respect to field $\Bbb Q$). Is $A$ dense in $\Bbb R?$
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4answers
326 views

Can a vector space have multiple spanning sets?

Maybe this is obvious, but can a vector space have multiple spanning sets or is there only a single spanning set for every vector space? Thanks
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1answer
200 views

How to show that two vector spaces $V$ and $W$ are the same

How to show that two vector spaces $V$ and $W$ are the same, if we know $\dim V = \dim W$ and $V$ is a subspace of $W$ ? Would it suffice to show there exists an isomorphism between them ? Any help ...
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1answer
15k views

Proof for triangle inequality for vectors

Generally,the length of the sum of two vectors is not equal to the sum of their lengths. To see this consider the vectors $u$ and $v$ as shown below. By considering $u$ and $v$ as two sides of a ...
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3answers
431 views

Does $\det(A) \neq 0$ (where A is the coefficient matrix) $\rightarrow$ a basis in vector spaces other than $R^{n}$?

I know that for a set of vectors $\{ v_{1}, v_{2}, \ldots , v_{n} \} \in \mathbb{R}^{n}$ we can show that the vectors form a basis in $\mathbb{R}^{n}$ if we show that the coefficient matrix $A$ has ...
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2answers
755 views

Realification and Complexification of vector spaces

I am interested in a good comprehensive resource on realification and complexification of vector spaces over the reals or complexes (and the interplay of these structures on the 'same' space in ...
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2answers
455 views

Is $(\mathbf{V} \cap \mathbf{W})^{\bot}=(\mathbf{V}^{\bot} \cap \mathbf{W}^{\bot})$?

Is $(\mathbf{V} \cap \mathbf{W})^{\bot}=(\mathbf{V}^{\bot} \cap \mathbf{W}^{\bot})$? I tried element-chasing, but I am getting confused when trying to determine mutual containment.
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2answers
3k views

How do you calculate the unit vector between two points?

I'm reading a paper on fluid dynamics and it references a unit vector between two particles i and j. I'm not clear what it means by a unit vector in this instance. How do I calculate the unit vector ...
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3answers
165 views

Is it possible to swap vectors into a basis to get a new basis?

Let $V$ be a vector space in $\mathbb{R}^3$. Assume we have a basis, $B = (b_1, b_2, b_3)$, that spans $V$. Now choose some $v \in V$ such that $v \ne 0$. Is is always possible to swap $v$ with a ...
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4answers
6k views

Linear Algebra: determine whether the sets span the same subspace

So I am stuck on 51 here: 51. Determine whether the sets $S_1$ and $S_2$ span the same subspace of $\mathbb{R}^3$: $$\begin{align*} S_1 &= \Bigl\{ (1,2,-1),\ (0,1,1),\ (2,5,-1)\Bigr\}\\ ...
4
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1answer
175 views

Normal $T\in B(H)$ has a nontrivial invariant subspace

I am wondering if the following is true: Every normal $T\in B(H)$ has a nontrivial invariant subspace if $\dim(H)>1$?
4
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2answers
346 views

A question about inner products on abstract vector spaces

I have been reading some materials and, for the n-th time in my life, there was a definition of an inner product as a function $V \times V \rightarrow F$, where $V$ is an abstract vector space and $F$ ...
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2answers
997 views

Showing that a set of trigonometric functions is linearly independent over $\mathbb{R}$

I would like to determine under what conditions on $k$ the set $$ \begin{align} A = &\{1,\cos(t),\sin(t), \\ &\quad \cos(t(1+k)),\sin(t(1+k)),\cos(t(1−k)),\sin(t(1−k)), \\ &\quad ...
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5answers
784 views

Subspaces of Hilbert Spaces of finite dimension

Given a Hilbert space $H$ of finite dimension, why is any subspace of this space closed? I tried bashing out an answer using an arbitrary Cauchy sequence $\{ f_1 , f_2, \ldots \} \subset S \subset H $ ...
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3answers
165 views

basis functions do not lie in the space they form

For example, any continuous function in $\mathbb{L}^2(-\infty,\infty)$ space can be expanded by delta functions $\delta(x-a)$ or Fourier basis $e^{ikx}$. However, the basis functions, both ...
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1answer
379 views

Center of Clifford Algebra depending on the parity of $\dim V$?

While reading about the structure of Clifford algebra, there were two facts listed as bullet points about the center of Clifford algebra based on the parity of the dimension of the underlying vector ...
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1answer
105 views

If the expectation $\langle v,Mv \rangle$ of an operator is $0$ for all $v$ is the operator $0$?

I ran into this a while back and convinced my self that it was true for all finite dimensional vector spaces with complex coefficients. My question is to what extent could I trust this result in the ...
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5answers
106 views

Minkowski sum of two disks

An open disk with radius $r$ centered at $\mathbf{p}$ is $D(\mathbf{p}, r)=\{\mathbf{q} \mid d(\mathbf p, \mathbf q) < r\}$, and the Minkowski sum of two sets $A$ and $B$ is $A \oplus B=\{\mathbf p ...
4
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1answer
102 views

To prove that the dimension of $V$ is $d_1^2 + \ldots + d_k^2$

Let $A$ be an $n \times n$ diagonal matrix with characteristic polynomial $$(x - c_1)^{d_1} \cdots (x - c_k)^{d_k} , $$ where $c_1,\ldots,c_k$ are distinct. Let $V$ be the space of $n \times n$ ...
4
votes
1answer
372 views

Splicing together Short Exact Sequences

Given an exact sequence of vector spaces $$\cdots\longrightarrow V_{i-1}\longrightarrow V_{i}\longrightarrow V_{i+1}\longrightarrow\cdots$$ I want to show that it is the same as having a collection of ...
4
votes
3answers
480 views

Why is the Kronecker delta a base for the dual vector space?

The Kronecker delta can be defined like this: $\delta_{ij} = \begin{cases} 1, & \text{if } i = j \\ 0, & \text{if } i \ne j \end{cases}$ The dual space is a vector space $V^*$ that can ...
4
votes
1answer
156 views

What does a subspace spanned by another subspace and a vector mean?

What does a subspace say A spanned by another subspace B and a vector x mean ? Does that imply anything about a basis or does it just mean that every vector in subspace A is either present in ...
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1answer
155 views

Is following subset W of V also a subspace?

Under following conditions $ a, b \in \mathbb{R}, V = \mathbb{R}^{2}, W = \{(x, y)\ |\ ax + by = 0 \} $ is W a subspace of V? I know the basics, but how would I prove that addition and ...
4
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1answer
128 views

Proof of $X\cup Y\neq V$

Suppose $X,Y$ are subspaces of dimension $n-k$ of the vector space $V$ of dimension $n$. Why is it always true that $X\cup Y\neq V$?  I can show this by arguing that if $X=Y$ then clearly by the ...
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2answers
240 views

Dot product of two vectors

How does one show that the dot product of two vectors is A · B = |A| * |B| * cos(Θ) ?
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4answers
489 views

Let $V$ be a $k$-vector space of dimension $n$ and $T: V \to V$ a linear map of rank 1. Show that either $T^2 = 0$ or that $T$ is diagonalisable?

I am not good with vector spaces so I would be grateful for any help. As I've been told I need to take $v \in \mathrm{Im}(T)$, $v\neq 0$, and show that if $T(v) = \mathbf{0}$ then $T^2 = 0$. But if ...
4
votes
1answer
62 views

Dual space and linear functional

The problem is this: Let $V$ be a vector space over $\mathbb{K}$ and $v \in V$. Show that if $f(v) = 0, \forall$ $f \in L(V, \mathbb{K})$, then $v=0$. It's a problem of a book I'm using to study Dual ...
4
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1answer
134 views

What does the symbol: $\mathcal{F}(S,F)$ in linear algebra mean?

I have a problem in linear algebra course, and I'm looking to solve it by myself, but I'm confused with notation since my teacher never mention it in class. It says: Let $S$ be a nonempty set and ...
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2answers
2k views

Vector space of polynomials

Do all polynomials $ax^3 + bx^2 + cx + d$ with a root at $x=1$ form a vector space? Do the coefficients $(a,b,c,d)$ form a vector space? My reasoning: Since $x=1$ is a root, we can't have $(a,b,c,d)$ ...
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2answers
185 views

Recasting points from one vector space to another

I have a collection of 3D points in the standard $x$, $y$, $z$ vector space. Now I pick one of the points $p$ as a new origin and two other points $a$ and $b$ such that $a - p$ and $b - p$ form two ...
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2answers
62 views

An example of space $V$ such that $(V^{\perp})^{\perp} \neq V$

I know that if $W$ is a vector space of finite dimension then for any subspace $V$ ,$(V^{\perp})^{\perp} = V$. But I have heard that this is not true for infinite dimensional vector spaces. So I tried ...
4
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1answer
32 views

$rk(A)=n$ implies $rk(AB)=rk(B)$

Let $A \in Mat_{m\times n}(\mathbb{R})$ and $B \in Mat_{n\times p}(\mathbb{R})$. Assume $rk(A)=n$. Prove that $rk(AB)=rk(B)$. Lets start by proving $rk(B) \ge rk(AB)$. Indeed, since the ...
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2answers
64 views

Show ker($\alpha$)=ker($\alpha$)^2 iff ker($\alpha$) and im($\alpha$) are disjoint

Let $V$ be a vector space over a field $F$ and let $\alpha$ be an element of $\operatorname{End}(V)$. Show $\ker(\alpha)=ker(\alpha^2)$ iff $\ker(\alpha)$ and $\operatorname{im}(\alpha)$ are disjoint. ...
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1answer
63 views

Left adjoint to direct sum?

In the category of vector spaces, is there some endofunctor $F$ satisfying $$\mathrm{Hom}_k(M,\underset{i \in I}{\bigoplus} k) \cong \mathrm{Hom}_k(F(M),k)$$ for every $k$-vector space $M$?
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2answers
118 views

Is there always an injective map from a space in its dual space?

Today our teacher said that dual spaces are "big" and told us that this is a consequence by Hahn-Banach's theorem. So I was wondering whether the dual space of a space is always "bigger" or equal ...
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2answers
63 views

Is $S=\{(1,t)\mid t\in \mathbb{R}\}$ a subspace of $\mathbb{R}^2$?

My professor introduced subspaces of $\mathbb{R}^n$ today and I don't think I understand them very well. He posed this question as an example: Is the set $S=\{(1,t)\mid t\in \mathbb{R}\}$ a ...
4
votes
3answers
163 views

Invertibility in a finite-dimensional inner product space

Let $T$ be an invertible linear operator on a finite-dimensional inner product space. I just want a hint as to how I should prove that $T^{\star}$ is also invertible and $( T^{-1} )^{\star} = ( ...
4
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2answers
60 views

Another linear algebra question

I have no idea how to start the following question. Any help will be greatly appreciated. (a) Let $A$ be a $n\times n$ matrix and let $a_1,...,a_n$ be the rows of $A.$ Suppose $y=(y_1, ..., y_n)$ is ...
4
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3answers
304 views

How do you construct the quaternion and the multiplication rules, like Hamilton did?

So, I understand complex number multiplication, and how it represents $2D$ rotations. What I don't understand is, how you add two more imaginary numbers $j$ and $k$, and get $3D$ rotations. I believe ...
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1answer
325 views

Image of unit ball dense under continuous map between banach spaces

I am assuming that the following problem will require the open mapping theorem, or maybe the closed graph theorem. Any help that can be given will be deeply appreciated. The statement is the ...
4
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2answers
117 views

Normed linear space and linear functional

Let $X$ be the normed linear spaceof sequences of reals that have only finitely many non-zero terms. Given $x = \{x_n\} \in X$, define $$f(x) = \displaystyle \sum_{n=1}^{\infty} x_n$$ I think that it ...
4
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1answer
234 views

Looking for proof that an open set in vector space contains the sum of two open sets.

Problem: To show that, in a topological vector space, for a given neighborhood of zero $W$, there exist two neighborhoods of zero, $V_1$, $V_2$, whose sum is contained in the first neighborhood, ...
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1answer
450 views

When are two diagonal matrices congruent?

This is probably a question that does not admit a simple answer. However, I'd like to know whether there exist criteria that determine when two diagonal matrices are congruent. I have the suspicion ...