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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Dimension of a subspace smaller than dimension of intersection

Suppose I have a finite-dimensional vector space $V$, and $U_1, U_2$ are subspaces of $V$, such that $U_1\nsubseteq U_2$. Is it possible that $\dim{U_1}\leq\dim{U_1 \cap U_2}$?
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How to convert from cartesian to vector form (of a straight line)

line in question: $$-x - 1 = \frac 12y - \frac12 = \frac 12z + 1$$ I wasn't really sure how to go about this one since it's not in the exact general form that I was taught, but I went along and ...
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1answer
47 views

What are the objects and morphisms of the category $\operatorname{Vect}$?

What are the objects and morphisms of the category $\operatorname{Vect}$? I am trying to learn category theory, and it seems we have infinite objects in $\operatorname{Vect}$ being all of the finite ...
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1answer
25 views

Vector inequality for a scalar difference of two vectors in $\mathbb{R}^n$.

A student posed an interesting problem to me the other day and embarrassingly I could not prove or disprove it even though it appears relatively simple. The question was: Given vectors $\mathbf{v},\...
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1answer
121 views

Finding normal basis for GF(q^m) over GF(q)

Could you kindly explain, how can one find a normal basis for GF$(3^6)$ over the GF$(3^2)$? As I understood, I should start with finding the polynomial in a form $$a(x^2) + (a^9)x + a^{81},$$ which ...
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3answers
322 views

Does $\{y\in \mathbb{R}^n:\operatorname{rank}((x,y,Ay))=2\}$ have zero Lebesgue measure?

This is probably a simple question, but I need some help. Consider a vector $x\in \mathbb{R}^n$ and a real $n\times n$ matrix $A$. I'm interested in the set of $y\in\mathbb{R}^n$ such that $x,y,Ay$ ...
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2answers
83 views

Basis 'vectors', basis 'matrices'?

Let $\mathfrak{sl}_2$ be the vector space of $2\times 2$ traceless matrices. Let $A\in \mathfrak{sl}_2$ be a diagonal matrix. Define a linear operator: $$\phi_A(X)=AX-XA$$ $\phi_A:sl_2\to sl_2$ What ...
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1answer
45 views

Null spaces associated to eigen values

Suppose that $f$ is an endomorphism in a finite dimensional vector space, and $\lambda$ is an eigenvalue of $f$. Let $C_f = \text{det}(f - X \,\text{Id})$ be the characteristic polynomial of $f$, and ...
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2answers
560 views

How to represent matrix as the sum of rank-one matrices

If we're given $B$ to be a $4 \times 7$ matrix: $$\begin{bmatrix}1 & 2 & -3 & 7 & 0 & -2 & 5\\1 & 2 & -3 & 7 & 1 & 3 & -2\\0 & 0 & 0 & 0 &...
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1answer
45 views

Dimension of a subspace of $\operatorname{Hom}(\mathbb{R}^3, \mathbb{R}^4)$

Let $V = \{f \in \operatorname{Hom}(\mathbb{R}^3,\mathbb{R}^4) \mid f(x,y,z)=f(z,x,y)\}.$ Find $\dim_\mathbb{R}(V)$. I think I need to use the dimension theorem for vector spaces: I need to ...
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0answers
83 views

How to calculate the dimension of an infinite direct product of copies of a field?

Let $F$ be a field and $I$ an arbitrary infinite index set. I'd like to know how to calculate the dimension of $\prod_{i\in I}F$. By the way, I know $\dim(\prod_{i\in I}F)\geqslant 2^{\text{card}(I)}...
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1answer
74 views

Prove that if the angle between any two of $k$ distinct vectors of Euclidean space $V$ is $\pi/3$ then $k \leq \dim V$

Prove that if the angle between any two of $k$ distinct vectors of Euclidean space $V$ is $\pi/3$ then $k \leq \dim V$. How I've approached this problem so far is to imagine it for $k = 2$ where if ...
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4answers
1k views

How to develop an intuitive feel for spaces

I'm a physicist who's currently delving deeper into what I would call more 'hardcore' maths (e.g. FEM and control theory). Every now and then, I come across various spaces, such as vector spaces, ...
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1answer
30 views

If $T:\mathbb{R}_3→\mathbb{R}_7$ is a linear transformation, then is the set $\operatorname{ker}(T)$ a subset of the codomain?

If $T:\mathbb{R}_3→\mathbb{R}_7$ is a linear transformation, then is the set $\operatorname{ker}(T)$ a subset of the codomain? My answer is 'yes', because $\operatorname{ker}(T) \subset \mathbb{R}_3 ...
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2answers
2k views

Basis for the null space of an identity matrix

Is the set containing only the zero vector a basis for the null space of an identity matrix?
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1answer
89 views

Linear dependence when number of vectors is greater than/less than the dimensions of the vector space

Simple question here, I just need some clarification of a theorem. Theorem: if k > n, then any k vectors in $R^n$ are linearly dependent. Nice and easy I guess! My question is this: Does this imply ...
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1answer
62 views

What is the probability of choosing r independent vectors in $\mathbb{R}^n$ in the unit sphere?

I was trying to compute the probability of choosing $r \leq n$ indepedent vectors $a_i \in \mathbb{R}^n$ such that they are independent. I was told that the probability that they are not independent ...
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1answer
31 views

Is this a vector space? If not, how can I make it one?

S (2x2 matrix) = {(a, b), (c,1) | $a,b,c$ is in $ \Bbb R$} I know for a vector space we must: 1. Define Addition 2. Define Scalar Multiplication 3. Have a set of numbers 4. Have a Field I know that ...
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3answers
84 views

What is abstraction of direction in considering vectors such as used in Engineering & Physics?

In the use of vectors of engineering and physics, we encounter objects that obey the axioms of a vector space but also have two new attributes of length (or, magnitude) and direction (e.g. direction ...
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1answer
86 views

Find a finite set of vectors which spans $W$.

Let $W$ be the set of all $(x_1, x_2, x_3, x_4, x_5)$ in $\Bbb R^5$ which satisfy $2x_1-x_2+{4 \over 3}x_3 - x_4\qquad = 0$, $x_1\qquad+{2 \over 3}x_3\qquad- x_5 = 0$, $9x_1-3x_2+6x_3-3x_4-3x_5 = 0$. ...
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1answer
32 views

Prove $V_{e} + V_{o} = V$

Prove $V_{e} + V_{o} = V$ where $V_{e}$ is a subset of even functions from $R$ into $R$, $V_{o}$ is a subset of odd functions from $R$ into $R$. I have proved $V_{e}$, $V_{o}$ are subspaces and $\...
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1answer
25 views

Show that the vectors form a basis for $R^3$.

Show that the vectors $\alpha_1 = (1, 0, 1)$, $\alpha_2 = (1, 2, 1)$, $\alpha_3 = (0, -3, 2)$ form a basis for $R^3$. Is it enough to show that the vectors are linearly independent?
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3answers
72 views

Gram-Schmidt Process and Orthogonal Components

Let the Gram-Schmidt process transform the vector system $(a_{1}, ..., a_{n})$ into the system $(b_{1}, ..., b_{n})$. Show that the vector $b_{k}$ is the orthogonal component of the vector $a_{k}$ ...
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1answer
65 views

Gram-Schmidt procedure on functions

I have been applying the Gram-Schmidt procedure with great success however i am having difficulty in the next step, applying it to polynomials. Here i what i understand If i have 2 functions, say $x^...
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0answers
125 views

What is the $\dim L(X,Y)$?

Let $X$ and $Y$ be two finite-dimensional vector spaces over the same field $K$, and let $L(X,Y)$ denote the vector space of all linear operators $T \colon X \to Y$. Then what is $\dim L(X,Y)$? My ...
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3answers
306 views

If $M, N$ are finite dimensional vector spaces with same dimension, then if $M$ is subset of $N$, then $M=N$.

If $M, N$ are finite dimensional vector spaces with same dimension, then if $M$ is subset of $N$, then $M=N$. I think i need to show that both vector spaces are spanned by the same bases in order to ...
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1answer
58 views

$\mathfrak{sp}_4$ is a subspace of the vector space of all $4\times 4$ matrices

Let $\mathfrak{sp}_4$ denote the set of all matrices $X$ satisfying $$X^TM+MX=0$$ How can I show that $\mathfrak{sp}_4$ is a vector subspace of the vector space of all $4\times 4$ matrices? I ...
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4answers
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Why is a function space considered to be a “vector” space when its elements are not vectors?

I am confused by the notion of a function space. For example consider the basis $\{1, x, x^2\}$ which is the basis for the vector space of all polynomials of degree at most $2$. What is the notion of ...
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1answer
325 views

Prove column space is a subspace of $\mathbb{R}^n$

I have an exercise on my last assignment for linear algebra, which is the following: The column space $C(A)$ of linear mapping $A: \mathbb{R}^m \rightarrow \mathbb{R}^n$ is defined by: $$C(A)...
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2answers
24 views

Prove $U = \{u(x) \in P_4 | u(0) + u(1) = 0\}$ is a subspace of $P_4$

I am trying to prove $U = \{u(x) \in P_4 | u(0) + u(1) = 0\}$ is a subspace of $P_4$ For U is nonempty I have: Let $u(x) = 0x^4 + 0x^3 + 0x^2 + 0x + 0$ For U is closed under $+$ I have: Let $x, y ...
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2answers
108 views

Why the column space of a matrix is useful?

I know what is the column space of a matrix: it is basically the subspace formed by the linear combinations of the columns (vectors) of a matrix. From wikipedia, we have the following nice picture: ...
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2answers
50 views

How to find if this huge vector is in the column space of this huge matrix?

I newbie to linear algebra, so I hope you are patient with me. I have to say if a vector $\vec{u} = \left[ \begin{matrix} 1 \\ 1 \\ 1 \\ 0 \\ 0 \\ 1 \\ 1\end{matrix} \right]$ is in the column space ...
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2answers
663 views

To prove any two basis of Finite Dimensional Vector Space have same number of elements

To prove any two basis of Finite Dimensional Vector Space have same number of elements If i take bases as $S_!$ = {$\alpha_!$ ,$\alpha_2$ ,....$\alpha_n$ } $S_2$ = {$\beta_!$,$\beta_2$ .... $\...
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2answers
24 views

existence of Matrix as an isomorphism

let $U$ be a subspace in $F^n$, let $V$ be a subspace in $F^m$, prove the existence of an $m\times n$ matrix $A$ such that $\text{Row}(A) = U$ and $\text{Col}(A) = V$ ok, um I forgot to mention ...
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1answer
24 views

Showing there exists an $m\times n$ matrix $A$ with $\text{Row}(A) = U$ and $\text{Col}(A) = V$ where $U$ and $V$ are subspaces.

Let $F$ be a field. Let $U$ be a subspace of $F^n$ and let $V$ be a subspace of $F^m$. Suppose that $\dim U = \dim V$. Then there exists an $m\times n$ matrix $A$ with entries in $F$ such that $\...
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0answers
62 views

Dimensions of spaces over different fields

We view $\Bbb C_2 = \{{w \choose z}:w,z\in \Bbb C\}$ as a vector space over $\Bbb C$, $\Bbb R$ and $\Bbb Q$. Let $x_1={i \choose 0}$, $x_2={\sqrt2 \choose \sqrt5}$, $x_3={0 \choose 1}$, $x_4={i\sqrt3 \...
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3answers
52 views

The only dense linear subspace of $\mathbb{C}^n$.

I want to prove this question: the only dense linear subspace of $\mathbb{C}^n$ is $\mathbb{C}^n$ itself. My immediat attempt is think of $\mathbb{C}^n$ as a closed subspace of itself, so it is its ...
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3answers
62 views

Commutativity of scalar/vector product: $a\mathbf{v}=\mathbf{v}a$ for all $a \in F$ and $\mathbf{v} \in V$

There are traditionally 8 axioms to check whether a set $V$ together with a field $F$ constitute a vector space. A common list of axioms can be found here. Missing from the list, however, is a ...
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1answer
47 views

How is this map injective?

Let $X$ be a (real or complex) vector space, let $X^{*}$ denote the vector space of all linear functionals defined on $X$, and let $X^{**}$ denote the vector space of all linear functionals defined on ...
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1answer
21 views

Find an orthogonal matrix such that its first line is $\frac{1}{5},\frac{2}{5}$

An orthogonal matrix is one matrix $A$ such that $A^t = A^{-1}$. So what I did: Suppose: $$A = \begin{bmatrix}\frac{1}{5}&\frac{2}{5}\\x&y\end{bmatrix}$$ Then: $$\begin{bmatrix}\frac{1}{5}&...
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1answer
31 views

Is it defined the product of vectors of different spaces?

I know that the sum of vectors of different spaces is not defined, but what about the multiplication of vectors of different spaces. For example, what about the multiplication of $v_1 = \left(\begin{...
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1answer
113 views

Find a isometry such that the matrix in respect to the canonical basis is:

I need to find a isometry such that the matrix in respect to the canonical basis is: $$\begin{bmatrix}\frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}& 0\\0 & 0 & 1\\x & y & z\end{...
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1answer
61 views

Complete the following proof that $-u$ is the unique vector in V such that $u+(-u)=0$.

suppose that $w$ satisfies $u+w=0$. Adding $-u$ to both sides we have $(-u)+[u+w]=(-u)+0$ $[(-u)+u]+w=(-u)+0$ $0+w=(-u)+0$ $w=-u$
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2answers
52 views

Orthogonal complement of subspace $W = span(5,1+t)$

I have this subspace of $P_2(\mathbb R)$ and I need to find its orthogonal complemente, using the inner product defined as $$<p(t),q(t)> = \int_o^1 p(t)q(t) dt$$ So I'm assuming the vector $$v =...
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1answer
41 views

Orthonormalization of basis ${1,1+t,2t^2}$ by $\langle u,v\rangle = \int_{0}^{1}uvdt$

Im doing the process: $$g_1 = \frac{1}{||1||} = 1\\v_2 = 1+t-<1+t,1>1 = 1+t-\int_{0}^{1}(1+t)1dt = t-\frac{1}{2}\\g_2 = \frac{v_2}{||v_2||}$$ but $$||v_2|| = \int_{0}^{1}t-\frac{1}{2}dt = 0$$ ...
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1answer
177 views

Basis for proper rational functions

Suppose $F$ is a field, and let $F(x)$ denote the $F$-vector space of all rational functions $\frac{f(x)}{g(x)}$, where $f,g\in F[x]$ are polynomials, with $g$ different from zero. Let $F(x)_p$ denote ...
0
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1answer
57 views

Is my proof about minimal polynomial correct

I am to prove that the characteristic polynomial and minimal polynomial have same roots. That is, if $\lambda$ is an eigenvalue of the linear transformation $T$ and if $p(t)$ is the minimal polynomial ...
1
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1answer
83 views

Closest Vector in a Inner Product Space

Let $V$ = $\mathbb{R}^n$ Note that $\langle -,-\rangle$ defines the Inner Product on $\mathbb{R}^n$ $$\|v\| = \sqrt{\langle v,v \rangle}$$ Consider the standard Distance Function $$d(x,y) = \|x-y\|...
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1answer
66 views

Clarification between a module and a vector space?

I'm reading Kenneth Hoffman's Linear Algebra, Ed2. In $\S5.5$ it talks about Module and Vector Spaces: (1) If $K$ is a commutative ring with identity, a module over $K$ ( or a $K$-module) is ...
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0answers
34 views

Do affine spaces have coordinate transformations?

I asked a question on Physics SE and there seemed to be some confusion as to whether affine spaces could have coordinate transformations. Specifically, the particular space I was working with was $\...