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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How do we show every linear transformation which is not bijective is the difference of bijective linear transforms?

I have been reviewing some ideas about vector spaces and came upon a surprising fact. I am not quite sure how to begin the argument because the problem requires one to construct two bijective linear ...
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448 views

Linear algebra - Memorising proper definitions of homomorphism types

I am reading a book about linear algebra. On the basis of this book, I worked out the terminology below. Problem: To me, it looks like Wikipedia defines homomorphism differently. Apart from that: Do ...
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2answers
74 views

$x,y$ are linearly depending iff $|\langle x,y\rangle|=||x|| \cdot ||y||$

I tried to prove a special case of Cauchy-Schwarz: $$x,y \text{ are linearly depending vectors} \Leftrightarrow |\langle x,y\rangle|=||x|| \cdot ||y||$$ $\Rightarrow$ is simple: \begin{eqnarray*} x= ...
6
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2answers
407 views

Is this proof of Cauchy Schwarz inequality circular or valid?

I'm a college freshman learning linear algebra on my own, and I'm in the section on inner products. I noticed a proof of the Cauchy Schwarz inequality for vectors in my book, and it seems to contain ...
6
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1answer
278 views

definition of ordered vector space

An ordered vector space is the pair $(V , \leq)$ where it satisfies the following: For all $x,y,z \in V, \lambda \geq 0$, i) $x \leq y \Rightarrow x+z \leq y+z$ ii) $x \leq y \Rightarrow \lambda ...
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1k views

Power-reduction formula

According to the Power-reduction formula, one can interchange between $\cos(x)^n$ and $\cos(nx)$ like the following: $$ \cos^n\theta = \frac{2}{2^n} \sum_{k=0}^{\frac{n-1}{2}} \binom{n}{k} ...
6
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2answers
492 views

examples of linear map $f:V \rightarrow V$, which is injective but not surjective

I am trying to find a linear map $f:V \rightarrow V$, which is injective but not surjective. I always thought that if the dimension of the domain and codomain are equal and the map is injective it ...
6
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3answers
2k views

Why orthogonal basis?

Lets take the $\mathbb{R}^3$ space as example. Any point in the $\mathbb{R}^3$ space can be represented by 3 linearly independent vectors that need not be orthogonal to each other. What is that ...
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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 ...
6
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1answer
188 views

Let $a_1, …,a_n , b_1,…b_n$ be $2n$ distinct elements of a field , then is the matrix $\Big(\dfrac1{a_i-b_j}\Big)_{ij}$ non-singular?

Let $a_1, ...,a_n , b_1,...b_n$ be $2n$ distinct elements of a field and define $$h_{ij}:=\dfrac1{a_i-b_j} , \forall i,j=1,2 ,\dots,n. $$ Is the $n \times n$ matrix $H:=(h_{ij})$ non-singular ?
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1answer
72 views

Series in a space which is not complete

Let $X$ be a normed vector space and $\left\lbrace x_n \right\rbrace_{n \in \mathbb{N}} \in X^{\mathbb{N}}$ with $$\sum_{n=1}^{\infty} \|x_n\| < \infty \wedge \sum_{n=1}^{\infty} x_n \notin X,$$ ...
6
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2answers
270 views

Vector spaces inquiry

Denote By $V$ the real vector spaces of all real polynomials in one variable, and let $P : V \rightarrow \mathbb{R}$ be a linear map. Suppose that $\forall$ $f,g \in V$ with $P(fg) = 0$ we have $P(f) ...
6
votes
2answers
113 views

Matrix ring $M_2(\mathbb{C})$, $\mathbb{C}^2$ with $M_2(\mathbb{C})$-module structure.

Let $R$ be the matrix ring $M_2(\mathbb{C})$. Let $M = \mathbb{C}^2$ with its natural $R$-module structure (just given by the usual action of $2 \times 2$ matrices on $2$-dimensional vectors). My ...
6
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3answers
320 views

Learning Math At Home

I want to learn math on my own at home. What is the best method to do so? I would say that I pick things up/grasp concepts pretty fast. I took math until grade 10 in highschool/secondary school and ...
6
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2answers
5k 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 ...
6
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1answer
397 views

What is the term for the projection of a vector onto the unit cube?

Normalizing a vector sets its magnitude to $1$ and retains its direction. In three dimensions, it projects the vector onto the unit sphere. Is there a term associated with projecting it onto the ...
6
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2answers
273 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
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2answers
578 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
405 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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1answer
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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
433 views

Positivity of the alternating sum associated to at most five subspaces

Let $V_1 , V_2 , \dots , V_n $ be vector subspaces of $ \mathbb{C}^m$ and let $$\alpha = \sum_{r=1}^n (-1)^{r+1} \sum_{ \ i_1 < i_2 < \cdots < i_r } \dim(V_{i_1} \cap \cdots \cap V_{i_r})$$ ...
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2answers
117 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 $\| ...
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250 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
votes
1answer
135 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
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1answer
375 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
227 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
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
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1answer
106 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
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1answer
98 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
422 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 ...
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1answer
2k 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
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3answers
189 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 ...
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3answers
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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
10k 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
71 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
42 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
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1answer
69 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
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1answer
105 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
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1answer
188 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 ...
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2answers
81 views

Decompose $V \otimes V \otimes V$ into irreducible representations of $SL_2(\mathbb{R})$

Let $V=\mathbb{C^2}$ be the standard representation of $SL_2(\mathbb{R})$ Decompose $V \otimes V \otimes V$ into irreducible representations of $SL_2(\mathbb{R})$ I will just consider ...
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64 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 ...
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48 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 ...
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1answer
158 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
311 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
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3answers
120 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
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2answers
642 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
221 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
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1answer
1k 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
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1answer
223 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
3answers
138 views

Why is it bad to pick basis for a vector space?

Reading `This Week's Finds', http://math.ucr.edu/home/baez/week247.html, I'm informed that one should avoid picking coordinate systems and I'm unsure why that is the case. Any help on the matter is ...