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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Finding a unit vector perpendicular to another vector

For example we have the vector $8i + 4j - 6k$, how can we find a unit vector perpendicular to this vector?
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2answers
355 views

Dimension of $\mathbb{Q}\otimes_{\mathbb{Z}} \mathbb{Q}$ as a vector space over $\mathbb{Q}$

The following problem was subject of examination that was taken place in June. The document is here. Problem 1 states: The tensor product $\mathbb{Q}\otimes_{\mathbb Z}\mathbb{Q}$ is a vector ...
4
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3answers
252 views

If $Ax=B$ has two solution, then there must be a third one?

How do I prove this conjecture? Let $A$ be a matrix, and $B$ be a column vectore. If $Ax=B$ has two solutions, then there must be a third one. Thanks in a advance!
4
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1answer
215 views

Is it possible to construct a quasi-vectorial space without an identity element?

I mean if there is any construction that satisfies all the conditions for an vectorial space except it lacks an identity element? This questions was posed to me by a classmate last semester and I have ...
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3answers
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Vector Spaces: Finding a basis and Dimension

I could really use some step-by-step help on these two problems please. Thank You in advance. 1.) Let $V = \{{\bf{A|A}}$ is an $n \times n$ matrix, $n$ fixed, det$({\bf{A}}) = 0$ }. Is $V$, with ...
4
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3answers
177 views

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
97 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 ...
4
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2answers
193 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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2answers
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 ...
4
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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
411 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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5answers
464 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
328 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
4
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1answer
204 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 ...
4
votes
1answer
16k 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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votes
3answers
439 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 ...
4
votes
2answers
767 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 ...
4
votes
2answers
458 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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votes
2answers
4k 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 ...
4
votes
3answers
166 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 ...
4
votes
1answer
181 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 $\dim(H)>1$?
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4answers
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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
votes
2answers
354 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
1k 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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votes
5answers
804 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 $ ...
4
votes
3answers
166 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 ...
4
votes
1answer
381 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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votes
2answers
54 views

Arbitrary (i.e. not necessarily finite-dimensional) vector spaces; reference request.

Its virtually impossible to complete an undergraduate degree these days without studying finite-dimensional vector spaces in quite some detail. So like most of us, I've done all that; however, just ...
4
votes
1answer
110 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 ...
4
votes
5answers
108 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
votes
1answer
107 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
375 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
503 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
157 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 ...
4
votes
1answer
156 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
votes
3answers
19k views

Find the equation of the plane passing through a point and a vector orthogonal

I have come across this question that I need a tip for. Find the equation (general form) of the plane passing through the point $P(3,1,6)$ that is orthogonal to the vector $v=(1,7,-2)$. I would ...
4
votes
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 ...
4
votes
2answers
242 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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votes
4answers
497 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
65 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
votes
1answer
137 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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votes
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 $(a,b,c,d)$ ...
4
votes
1answer
7k views

Average of multiple vectors

I have more than two vectors (these are 2D vectors) and I want to calculate the mean vector. What is the correct way to do it? All my vectors share their origins at (0,0).
4
votes
2answers
186 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 ...
4
votes
1answer
199 views

Null space basis

Let $V\in\mathbb{R}^{a\times b}$ be a matrix such that it is not a full column rank. Then there will be a nonsingular matrix $H$ such that $$VH=\left[\begin{array}{cc} V_{1} & 0_{a\times ...
4
votes
2answers
66 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
votes
1answer
33 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 ...
4
votes
2answers
65 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. ...
4
votes
1answer
64 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 $M$?