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 ...

learn more… | top users | synonyms

1
vote
0answers
24 views

Vector space complement to a multiplicatively closed subspace is an ideal

Let $V$ be a vector space over $\mathbb{C}$ of any dimension and suppose we have an associative multiplication $V \times V \to V$ making $V$ into a commutative ring with unity. Let $V=U \oplus W$ be a ...
2
votes
4answers
410 views

A Vector Space is a Set - Axiom or Derivation?

I understand that structures with the properties of the real and complex numbers can be defined and derived from the axioms of ZFC set theory. But can a structure with the properties of a (possibly ...
3
votes
3answers
121 views

Can someone explain this: “the set of subspaces of a vector space ordered by inclusion”

This is a claim on Wikipedia https://en.wikipedia.org/wiki/Partially_ordered_set I am not sure how to make sense of the claim What does it mean by ordered by inclusion? Inclusion as in $\subseteq$? ...
-3
votes
0answers
45 views
0
votes
1answer
12 views

The map $f$ is degenerate or non-degenerate?

Let denote by $M_{3,2}(\mathbb C) $ the space of all $(3\times2)$-matrix of complex-dimension equal $6$ with basis $(E_{1},E_{2},E_{3},E_{4},E_{5},E_{6})$. Let $f$ a $\mathbb R$-bilinear skew-...
0
votes
1answer
26 views

Solving vector equation 2

Using vector method, show that the vector equation $$\bar{x}\times \bar{a}+(\bar{x}.\bar{b})\bar{c}=\bar{d}$$ is satisfied if $$\bar{x}=\lambda \bar{a}+\bar{a}\times \frac{\bar{a}\times (\bar{d}\...
0
votes
2answers
27 views

Solving vector equation 3

Solve for $\bar{x}$ and $\bar{y}$ $$\bar{x}+\bar{y}=\bar{a},~~ \bar{x}\times \bar{y}=\bar{b},~~ \bar{x}.\bar{a}=1$$ Attempt: $\bar{x}+\bar{y}=\bar{a}$ dot by $\bar{a}$, we get $1+\bar{a}.\bar{y}=|...
2
votes
1answer
40 views

Dual basis vectors and Basis one-forms

I'm studying Tensor Calculus on some MIT's notes (page 16) and I'm stuck at the point where it defines dual basis vectors. I have already studied basis one forms and I can't understand why we need to ...
0
votes
1answer
17 views

Rotation of 3d vector alone a plane?

I have vector PQ which lies on plane Ax+By+Cz+D=0, now after i rotate this vector in this plane with angle t,about the point P what will be the new position of Q ? Here the position of new Q is ...
3
votes
2answers
99 views

Motivation behind word quotient

Why set of all cosets of a subspace W of a vector space V is called quotient space. What is the motivation behind word quotient?
1
vote
3answers
52 views

Every subspace is the kernel of a linear map

I know that every kernel of a linear map from $\mathbb R^n$ to $\mathbb R^m$ is a subspace of $\mathbb R^n$. I am wondering if the converse is true, i.e. every subspace of $\mathbb R^n$ is the kernel ...
0
votes
1answer
23 views

How can find the vector that satisfy some conditions

I have a question Assume that there are 3 vectors x1,x2,x3 (each vector has the size 3*1 (3 dimension)) I want to find these vector that satisfy below conditions (the ininitial assumption x1 = [1 0 ...
3
votes
2answers
424 views

Vector Project onto Subspace

So the question is: Let S be the subspace of $\mathbb{R}^3$ spanned by the vectors $ u_2 = \begin{pmatrix} \frac{2}{3}\\\frac{2}{3}\\\frac{1}{3}\end{pmatrix} u_3 = \begin{pmatrix} \frac{1}{\sqrt{2}}\\\...
0
votes
1answer
18 views

element-wise order implies norm order?

Let $v_1, v_2 \in \mathbb R^n$. If $0\le v_1 \le v_2$ element-wise, is it true that $\|v_1\| \le \|v_2\|$ for any norm $\|\cdot \|$?
4
votes
2answers
158 views

What are some usual norms for matrices?

I am familiar with norms on vectors and functions, but do there exist norms for spaces of matrices i.e. $A$ some $n \times m$ matrix? If so, that would that imply matrices also form some sort of ...
1
vote
1answer
21 views

Proof of: a vector space spanned by $r$ vectors has dimension $\leq r$

I am confused about this proof of this statement in baby Rudin (Theorem 9.2 in third edition pp. 205). If a vector space $X$ is spanned by r vectors, then dimension($X$)$\leq r$ The proof goes ...
2
votes
3answers
263 views

What is the product and coproduct of Morphism category (Arrow category)?

Given a category C, its morphism category D means a category that has 1) "morphisms of C" as its objects 2) "pairs (f,g) s.t. the diagram (square) commutes" as its morphisms. The above definition ...
1
vote
3answers
144 views

Does $K = \mathbb Q[X]/(X^4 - 2)$ contain the imaginary unit $i$?

Let $P(X) = X^4 - 2 \in \mathbb Q[X]$. a) Prove that $P(X)$ is irreducible. b) Prove that the field $K = \mathbb Q[X]/(P(X))$ is an algebraic extension of $\mathbb Q$ and find a generator of it....
2
votes
1answer
602 views

Triangle and parametric coordinates

I'm studying on a book where it says: "A triangle is the set of points where for some point po, where u and v range over the parametric coordinates (we are talking about barycentric coordinates ...
1
vote
1answer
43 views

If $Ax = O$ has only one solutions, the columns of A span R?

I've been doing some excersices about inner product and I found something interesting but I don't know if my approach is correct at all. Supose that ${v_{1}, v_{2}, ..., v_{n}}$ is a base for a ...
4
votes
3answers
398 views

A vector space over an infinite field is not a finite union of proper subspaces? [duplicate]

Show that if $V$ is a vector space over an infinite field $\mathbb{F}$, then $V$ cannot be written as set-theoretic union of a finite number of proper subspaces.
0
votes
2answers
47 views

How to describe range of a linear transformation?

I'm self studying Linear Algebra from Hoffman Kunze, and I've come upon this problem. With complex number $z=x+iy$, $$T(z)=\begin{pmatrix} x-7y & 5y \\ -10y & x+7y \\ \end{pmatrix}$$ is ...
0
votes
3answers
46 views

Determine if the following vectors are coplanar.

I have no idea to start with this question, I know how to find if vectors are coplanar when the values of the vectors are given to me, but I do not know how to manipulate coplanarity properties well ...
0
votes
0answers
20 views

Dimension of cartesian product of two vector space [on hold]

If V and W are vector spaces over F then what is dimension of vector space V×W over F
2
votes
2answers
110 views

Proving that a space is complete

There is something that bugs me about the proof I've been shown that $C(\Omega)$ (the space of continuos function on $\Omega$, a compact subset of $\mathbb R^n$) with the $\sup$ norm is complete. ...
0
votes
0answers
66 views

Prove that N(T)=0 and R(S)=U

Let $T:U \to V$; $S:V \to U$ and $ST:U \to U$. Prove that $N(T)=\{0\}$ and $R(S)=U$. My professor gave us a fact at some point that if $ST=ID(U)$ we have S is surjective and T is injective. I am not ...
0
votes
0answers
9 views

SVM how is the quaradic problem set up?

I am at the moment trying to understand svm and, are having some troubles undetrstanding why SVM tries to find two closest point of the 2 convex hulls as explained in the paper: Support Vector ...
1
vote
1answer
34 views

What is the intuition behind Gramian method for linear independence? and Is there $simple$ proof of it?

I'm trying to figure out the intuition behind Gramian method to determine the linear independence of functions. I searched the web for such simple intuitive explanation and found nothing. I tried ...
2
votes
1answer
51 views

Let $X = C[a,b]$ be a vector space of continuous real functions on the interval $[a,b] \subset{R}$ and $\|x\| = \sup\{|x(t) : t \in [a,b] \}$

Let $X = C[a,b]$ be a vector space of continuous real functions on the interval $[a,b] \subset{R}$ and $\|x\| = \sup\{|x(t) : t \in [a,b] \}$, $x\in X$, norm on $X$. Prove that with $(Ax)t = t^2x(a)$, ...
0
votes
3answers
44 views

Basis for a subspace

I need to calculate the basis for $$W = \lbrace (a,b,c,d) \: : \: a+b+c = 0 \rbrace.$$ I find it hard to understand how does the fact that d is not part of the equation effects the basis. Thanks ...
2
votes
2answers
41 views

Diagonalizable by orthonormal matrix

Given the matrix $$A = \begin{bmatrix} 2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2 \end{bmatrix}$$ Explain why $A$ can be diagonalized by an orthonormal matrix and find an ...
2
votes
2answers
526 views

Space spanned by matrices

I have a set of $5$ by $5 $matrices, $M_1,M_2,...,M_{19} ,M_{20}$. I want to try to find a basis from this set and also to find relationships between these matrices. This is how I think I should ...
2
votes
2answers
43 views

Show that for each $n \in \mathbb{N}$, $\operatorname{span}\{p_1(x),\ldots,p_n(x)\} = \operatorname{span}\{1,x,x^2,\ldots,x^n\}$

Assume that, for each $n \in \mathbb{N}$, we have $p_n(x)=\sum_{k=0}^{n-1} x^k$ . Show that for each $n \in \mathbb{N}$, $$\operatorname{span}\{p_1(x),\ldots,p_n(x)\} = \operatorname{span}\{1,x,x^2,...
4
votes
0answers
46 views

On kernels of commuting operators in infinite dimensions

Let $X$ be an infinite dimensional vector space, and let $\operatorname{S},\operatorname{T}\in\mbox{End}(X)$ be two operators such that: $\operatorname{T}\operatorname{S}=\operatorname{S}\...
1
vote
2answers
32 views

Determining a basis for a space of polynomials.

Let $V = \mathbb R[x]_{\le 3}$ I have the space of polynomials $U_2 = \{ p = a_0 + a_1x + a_2x^2 + a_3x^3 \in V \mid a_1 - a_2 + a_3 = 0, a_0 = a_1 \}$ I am asked to find a basis, so I proceed by ...
1
vote
1answer
34 views

On the dimension of subspaces of the vector space given by the product of polynomials.

I was asked this question orally so feel free to also correct how the question is written. Given the vector space of polynomials in the variable $x$ with degree $\le 4$ and the vector space of ...
0
votes
1answer
1k views

Find the vector equation of a line passing through point A perpendicular to line AB

'Points A and B have coordinates (4,1) and (2,-5) respectively. Find a vector equation for the line which passes through the point A (only the point A), and which is perpendicular to the line AB.' ...
2
votes
3answers
85 views

Why can't a set of four vectors in $\mathbb{R}^3$ be linearly independent?

Why can't a set of four vectors in $\mathbb{R}^3$ be linearly independent? I know that if the determinant of the vectors together is not $0$ then the vectors are linearly independent. But this is ...
1
vote
3answers
54 views

Method of Proof in Showing Something is Smallest (Subspace)

I am reading a proof that shows the sum of subspaces is the smallest subpsace containing all the summands (It is a vector space over $\mathbb{R^n}$). The author of the book goes to show first it is a ...
1
vote
1answer
24 views

Collinear Points in 3-Dimensions

The points A(3, -1, z), B(1, 2, 6), and C(x, 8, 14) are collinear. Find the values of x and z. I have tried finding common ratios between the points, but no common ratio is possible, I have a feeling ...
2
votes
1answer
39 views

Parametrized linear operator

I've been trying to solve the following task: Determine $a$, $b$ $\in \mathbb{R}$ so that for the linear mapping $A :\mathbb{R}^3\to\mathbb{R}^3 $, with linear transformation matrix $$\mathcal{M}(...
0
votes
3answers
45 views

Orthogonal complement and projection

Let $M$ be a subspace of $\mathbb R^4$ which is spanned by the vectors $v_1 = (1,0,-1,1)$ , $v_2=(0,1,2,1)$. Find the orthogonal complement $M^T$ of $M$ and the orthogonal projections of the vector $v=...
3
votes
1answer
40 views

Precedence of operations in vector spaces

Suppose that $V$ is a vector space over $\mathbb R$ (for simplicity) with addition denoted by $\oplus$ and scalar multiplication denoted by $\otimes$. Let $\mathbf u, \mathbf w \in V$ and let $\lambda ...
0
votes
1answer
29 views

Show that $\operatorname{Span}(C) = V_1 \cap V_2$

Let $\mathbb{R}[x]$ be the set of polynomials, and let $$V_1 = \{a_1x + a_2x^3 + a_3x^5 \mid a_1, a_2, a_3 \in \mathbb{R}\}$$ $$V_2 = \{b_1x^2 + b_2x^3 + b_3x^4 \mid b_1, b_2, b_3 \in \...
3
votes
1answer
5k views

How to find perpendicular distance from point to plane in $3D$.

The line $L_1$ passes through point $A$ whose position vector is $3i - 5j + 4k$, and is parallel to the vector $3i + 4j + 2k$. The line $L_2$ passes through the point $B$ whose position vector is $2i +...
1
vote
2answers
470 views

Angle between two vectors not in same plane

I want to know how calculate the angle between two vectors and both are not in same plane, which means that they don't intersect at any point? For example how do I calculate angle between AB and EF ...
1
vote
1answer
23 views

Prove that $T(u)$ is linearly independent in $W$

Let $V$ and $W$ be two vector spaces over $\mathbb{R}$ Suppose $X \subseteq V$ is a nonempty linearly independent set and $T:V \rightarrow W$ is an injective linear map. Prove that {$T(u): u \in X$} ...
1
vote
1answer
32 views

Linear Transformation Basis Exercise

I have tried to solve the following exercise. Is it right? Consider the linear transformation L: ℝ⁴→ ℝ³. Knowing that: $$ L \begin{pmatrix}2\\0\\0\\0\end{pmatrix} = \begin{pmatrix}2\\2\\2\end{...