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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244 views

Cross products?

Say you have vectors $v$ and $w$. Let there cross product be denoted by $x$ so that: $$v \times w = x$$ According to Wikipedia: $$x_x = v_yw_z - v_zw_y$$ $$x_y = v_zw_x - v_xw_z$$ $$x_z = v_xw_y - ...
6
votes
2answers
502 views

Probability that $n$ vectors drawn randomly from $\mathbb{R}^n$ are linearly independent

Let's take $n$ vectors in $\mathbb{R}^n$ at random. What is the probability that these vectors are linearly independent? (i.e. they form a basis of $\mathbb{R}^n$) (of course the problem is ...
6
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1answer
332 views

Is there a difference between abstract vector spaces and vector spaces?

I am following my Oxford syllabus and my next step is abstract vector spaces, in my linear algebra book I've found vector spaces. I've searched a little and made a superficial comparison between ...
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2answers
1k views

Does anyone know any resources for Quaternions for truly understanding them?

I've been studying Quaternions for a week, on my own. I've learned various facts about them but I still don't understand them. My goal is to understand rotation quaternions specifically. I don't want ...
6
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2answers
4k views

Relation between cross-product and outer product

If inner products ($V$) are generalisations of dot products ($ \mathbb{R}^n$), then are outer products ($V$) also related to cross-products ($ \mathbb{R}^3$) in some way? A quick search reveals that ...
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1answer
1k views

direct products and direct sums for matrices and for vector spaces

I was wondering what relations and similarities are between direct product for matrices and direct product for vector spaces? Or do they just unfortunately and somehow misleadingly happen to have the ...
6
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2answers
104 views

What is the implication that $\| \cdot \|_2$ and $\| \cdot \|_\infty$ are equivalent norms on $\mathbb{R^2}$

Given $\mathbb{X}$ = $\mathbb{R^2}$, consider $\| \cdot \|_2$ and $\| \cdot \|_\infty$ We can show that $\| x \|_\infty \leq \| x \|_2 \leq \sqrt2 \| x \|_\infty$ Hence $\| \cdot \|_2$ and $\| ...
6
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2answers
239 views

Question about Normed vector space.

Here is the definition of a normed vector space my book uses: And here is a remark I do not understand: I do not understand that a sequence can converge to a vector in one norm, and not the ...
6
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1answer
1k views

Dimension of the vector space of homogeneous polynomials

Let $k[X_0, X_1, \ldots, X_n]_d$, or briefly $k[X]_d$, be the $k$-vector space whose elements are the zero polynomial and homogeneous polynomials of degree $d\geq 1$. I found the following formula for ...
6
votes
1answer
129 views

Linear and Commutative function over Square Matrices.

Find all functions $f$, such that $f(mA+nB) = mf(A) + nf(B)$ and $f(AB) = f(BA)$ , where $A, B$ are square matrices and $ m,n$ are scalars. Need to find $f$ as an explicit function of any general ...
6
votes
1answer
346 views

Proof of a direct sum decomposition

I was trying to prove this statement: If $N: V \to V$ is a nilpotent operator on a complex vector space, $N^k=0$ and $U\subset V$ is a subspace with $U \cap \ker(N^{k-1})= \{0\}$ then there exists ...
6
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1answer
217 views

Alternative Almost Complex Structures

Let $V$ be a real vector space. An almost complex structure on $V$ is a map $J : V \to V$ such that $J^2 = -\mathrm{id}_V$. An almost complex structure gives $V$ the structure of a complex vector ...
6
votes
1answer
4k views

Why are vector spaces sometimes called linear spaces?

I have never come across the term 'linear space' as a synonym for 'vector space' and it seems from the book I am using (Linear Algebra by Kostrikin and Manin) that the term linear space is more ...
6
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1answer
1k views

Cross product of vectors as a determinant: valid matrix operation?

"The definition of the cross product can also be represented by the determinant of a formal matrix." —Wikipedia This seems like a hack to me—something of much practical use but ...
6
votes
1answer
85 views

Why are inner product spaces only defined on $\Bbb R$ or $\Bbb C$?

A vector space $V$ makes sense over any field $F$, or even a division ring. So why does adding an inner product suddenly not make sense without taking the $F=\Bbb R$ or $\Bbb C$? What are the primary ...
6
votes
1answer
83 views

Given a vector space with two inner products, there is a linear transformation taking one to another

I am looking for some hint to the following question: Let $V$ be an $n$-dimensional real inner product space and let $\langle x,y\rangle$ and $[x,y] $ both be two different inner products on V. ...
6
votes
1answer
199 views

Duality mappings on finite-dimensional spaces

I have a few questions regarding some concepts in a book "nonlinear partial differential equations by Roubicek" I am studying. The following is from the text. "Let $V$ be a separable, reflexive ...
6
votes
1answer
547 views

How much can we “cheat” and use vector knowledge in complex analysis?

I'm an engineering-physics student taking a course in complex analysis, and it's a little frustrating, because I see all these connections to vector calculus over the reals (especially as applied to ...
6
votes
3answers
179 views

Do all vectors belong to a vector space?

If we were given the vector $(1,1,1)$, say, we know immediately that it belongs to the vector space $\mathbb{R}^3$ (and infinitely many others). But, if we take this vector of dolphins: or some ...
6
votes
3answers
2k views

Question on finite Vector Spaces, injective, surjective and if $V$ is not finite

Let $V$ be a vector space and $\alpha \in \operatorname{End}(V)$ (i) If $V$ is finite dimensional, then $\alpha$ is injective iff $\alpha$ is surjective. (ii) Give example showing (i) is false if ...
6
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1answer
8k views

Properties of a matrix whose row vectors are dependent

When a column vector in a matrix is a made up of "combination" of its other column vectors, it is said to be linearly dependant. Say... $$ A=\begin{bmatrix} 2 & 1 & 0\\ 4 & 5 & ...
6
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1answer
65 views

$SL_2(\mathbb{F})$, decomposing $\mathbb{C}\{X\}$ into irreducible $G$-representations and dimensions

Let $\mathbb{F}$ be a finite field with $q$ elements and $H = \mathbb{F}^\times$, the multiplicative group of $\mathbb{F}$. It is known that $H$ is a cyclic group of order $q - 1$, so $\widehat{H} = ...
6
votes
1answer
32 views

How to show the sum of the images of such $m$ projections is direct and is the whole space?

There are $m$ projections (whose square are themselves) $\phi_1,\cdots,\phi_m$ acting on a finite-dimensional vector space $V$ such that $$\phi_i\phi_j=0\quad i\ne j\tag{1}$$ where $0$ denotes the ...
6
votes
1answer
68 views

Closed formula for Poincaré series in terms of adjacency matrix.

Let $Q$ be a finite quiver with vertex set $I$. For each $n = 0, 1, 2, \dots,$ let $k^{(n)}Q \subset kQ$ be the $k$-linear span of all paths of length $n$, in particular, we have$$k^{(0)}Q = ...
6
votes
1answer
104 views

Maximal value of dimension [closed]

I'm stuck on a question if you can help me : Show that the maximum dimension of a subspace of $\mathcal M_n (\mathbb F)$ not containing an invertible matrix is $n (n-1)$.
6
votes
1answer
186 views

Clarification: Viewing $\mathbb{R}^n$ as a probabilistic state space

In this MathOverflow post on visualizing high-dimensional spaces, Terry Tao states that "the fact that most of the mass of a unit ball in high dimensions lurks near the boundary of the ball can be ...
6
votes
2answers
62 views

Uses of vector spaces over $\mathbb Q$

I know of two applications of vector spaces over $\mathbb Q$ to problems posed by people not specifically interested in vector spaces over $\mathbb Q$: Hilbert's third problem; and The Buckingham pi ...
6
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2answers
39 views

Linear functions and intersections of null subspaces

Let $V$ be a vector space of a finite dimension $n$ over the field $K$. Let $\phi, \psi$ be two non-zero functionals on $V$. Assume that there is no non-zero element $c \in K$ such that $\psi= c ...
6
votes
1answer
146 views

what is vector $(\vec{a}\cdot \vec{b})\vec{c} + (\vec{b}\cdot \vec{c})\vec{a} - (\vec{c} \cdot \vec{a})\vec{b}$

Suppose we have three non orthogonal vectors in $R^3$ as $\vec{a}, \vec{b}, \vec{c}$. The vector of $(\vec{b}\cdot \vec{c})\vec{a} - (\vec{c} \cdot \vec{a})\vec{b}$ is in the plane spanned by ...
6
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2answers
290 views

Why do we need dual space [closed]

In functional analysis there are many places where dual space is mentioned, but I still don't understand the real power of that concept. Why do we need the dual space?
6
votes
3answers
116 views

What are some uses for other norms on $\mathbb{R}^n$

We all know and love the standard $1,2,$ and $\infty$-norms on $\mathbb{R}^n$. However, I have never seen anyone mention uses for any of the other $k$-norms that I'm defining as ...
6
votes
2answers
593 views

Complement of all-one vector in binary vector space

Let $V$ be a k-dimensional subspace of $(\mathbb{F}_2)^n$, such that vector $\vec{j}=(1,1,...,1) \in V$. Standard linear algebra shows that it is possible to find a $(k-1)$-dimensional space $W$ such ...
6
votes
1answer
208 views

How many parameters are required to specify a linear subspace?

A problem in Peter Lax's Linear Algebra involves looking at the family of $n\times n$ self-adjoint complex matrices and asking: on how many real parameters does the choice of such a matrix depend? ...
6
votes
1answer
968 views

dimension of the space of all symmetric matrices with trace $0$ and $a_{11}=0$,

I want to know the dimension of the space of all symmetric matrices with trace $0$ and $a_{11}=0$, I can show that the dimension of space of all symmetric matrices $S$ is $n(n+1)/2$, now I give a ...
6
votes
1answer
295 views

Calculating the intersection of two spaces of polynomials

This problem is driving me nuts. I feel like there should be an elementary argument, yet I have failed to find one. Consider the vector space $V_n=\mathbb Q[x]/{x^{2n+1}}=\mathbb Q\{1,x,x^2,\ldots, ...
6
votes
1answer
281 views

General Steinitz exchange lemma

Where can I find a proof of the following general Steinitz exchange lemma: Let $B$ be a basis of a vector space $V$, and $L\subset V$ be linearly independent. Then there is an injection ...
6
votes
1answer
199 views

Make a vector space to “house” a parabola

I've had this idea I find interesting. A line on a plane or in space that goes through the origin of the system is a vector space because if you add up or multiply with a scalar any of it's elements, ...
6
votes
1answer
218 views

When does there exist an isometry that switches two subspaces?

Let $V$ be a real vector space of finite dimension and let $\langle \cdot, \cdot \rangle$ be a non-degenerate symmetric bilinear form on $V$. Let $U, W \subseteq V$ be linear subspaces such that ...
6
votes
1answer
97 views

Invariant Subspace of Two Operators [duplicate]

Let $S$, $T$ be linear operators on a finite-dimensional vector space $V$ over $\mathbb{C}$. Suppose $$S^2 = T^2 = I.$$ Show that there exists either a $1$-dimensional or $2$-dimensional ...
6
votes
0answers
89 views

When are all ring homomorphisms also algebra homomorphisms?

Let $k$ be an algebraically closed field, and let $A,B$ be two unitary $k$-algebras. In general, there are more ring homomorphisms $A\to B$ than there are $k$-algebra homomorphisms. More precisely, ...
6
votes
0answers
414 views

Finding the maximum number of subspaces of a vector space over finite field that satisfy these relations

I have a question and I am stuck. I was wondering if anyone has a thought, before I start a brute-force search. For $q$ a prime number and $n =6$, let $\mathbb {F}_{q}^{n}$ be an $n$-dimensional ...
6
votes
2answers
81 views

$C(M)=\{A\in M_n(\mathbb{C}) \mid AM=MA\}$ is a subspace of dimension at least $n$.

Let $M_n(\mathbb{C})$ denote the vector space over $\mathbb{C}$ of all $n\times n$ complex matrices. Prove that if $M$ is a complex $n\times n$ matrix then $C(M)=\{A\in M_n(\mathbb{C}) \mid ...
5
votes
8answers
1k views

Is it too much rigor to turn a set into a vector space?

I was reading some online notes on vector spaces and one authors insisted on turning a set $\mathbb{X}$ into a vector space. I thought it was quite insane but maybe I am not seeing the point. The ...
5
votes
6answers
667 views

Is there such thing as an unnormed vector space?

I learned about Banach spaces a few weeks ago. A Banach space is a complete normed vector space. This of course made me wonder: are there unnormed vector spaces? If there are, can anyone please ...
5
votes
3answers
553 views

Why do we use n-dimensional spaces?

On mathoverflow, Terry Tao says the following: For instance, one can view a high-dimensional vector space as a state space for a system with many degrees of freedom. A megapixel image, for instance, ...
5
votes
4answers
2k views

What are some alternative definitions of vector addition and scalar multiplication?

While teaching the concept of vector spaces, my professor mentioned that addition and multiplication aren't necessarily what we normally call addition and multiplication, but any other function that ...
5
votes
3answers
696 views

Why a non-diagonalizable matrix can be approximated by an infinite sequence of diagonalizable matrices?

It is known that any non-diagonalizable matrix, $A$, can be approximated by a set of diagonalizable matrices, e.g. $A \simeq \lim_{k \rightarrow \infty} A_k$. Why this is true? Note: I was faced with ...
5
votes
2answers
2k views

Proving that $\mathbb{F}^\infty$ is infinite-dimensional.

I'm supposed to prove that $\mathbb{F}^\infty$ is infinite-dimensional. I was planning on doing a proof by induction to show that $(1,0,...),(0,1,0,...),...$ is a basis. Is this permissible? Also, I ...
5
votes
4answers
152 views

Nullspace that spans $\mathbb{R}^n$?

My professor said that if for a $n \times n$ matrix $A$, $\text{null}(A) = \mathbb{R}^n$, then $A = 0_{n}$. Why is this true? I understand what its saying - if everything times this matrix is zero, ...
5
votes
3answers
393 views

Every invertible linear transformation can be perturbed a bit without destroying invertbility, Neumann series

Let $T: V \to V$ be any linear transformation on a real or complex vector space $V$. Show that there exists $\epsilon_0 > 0$ $($depending on $T$$)$ so that $I + \epsilon T$ is invertible for any ...