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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1answer
40 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
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3answers
393 views

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

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.
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3answers
118 views

Does $K = \mathbb Q/(X^4 - 2)$ 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 $K = \mathbb Q/(P(X))$ is an algebraic extension of $\mathbb Q$ and find a generator of it. ...
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2answers
43 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 ...
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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 ...
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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
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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. ...
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0answers
62 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 ...
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0answers
7 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
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1answer
32 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
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1answer
49 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 $(Ax)t = t^2x(a)$, ...
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3answers
43 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
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2answers
40 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
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2answers
525 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
42 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,...
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0answers
44 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}\...
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vote
2answers
30 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
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1answer
33 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 ...
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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 ...
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3answers
52 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
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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}(...
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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
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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
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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 +...
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2answers
469 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
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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
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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{...
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1answer
23 views

Linear transformation explanation

I have the following exercise: Consider the linear transformation L: ℝ³→ ℝ². Knowing that: $$ L \begin{pmatrix}1\\1\\0\end{pmatrix} = \begin{pmatrix}1\\2\end{pmatrix} \space\space\space\space\...
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0answers
18 views

Prove that $(f, g, h)$ is a linearly independent list of vectors in $\mathbb{R}[x]^S$

"Recall that $\mathbb{R}[x]$ is a vector space. Suppose that $f, g, h \in \mathbb{R}[x]^S$ and that there is $q \in S$ such that $f(q) = 1$, $g(q) = x^2 + 1$, and $h(q) = x^2 + x$. Prove that $(f, g, ...
1
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1answer
21 views

Understanding components of a vector

I learned that we can get the component of a vector in any direction using the dot product. The problem I have is the meaning of the term component itself. The component of a vector $\vec A$ in the ...
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0answers
18 views

Is it better to average the log2 for a series of numbers or just the numbers themselves? And, how would you test or prove this?

Lets say I'm trying to compare two vectors for similarity and normalizing them before hand based on some mean or standard deviation combo for the purpose of finding the similarity between the 2 ...
0
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1answer
56 views

Differentiation w.r.t. the $\mbox{vec}$ operator

I am stuck at solving the following derivative $$\frac{d \mbox{vec} (X^T X)}{d \mbox{vec} (X)}$$ where $X$ is an $m \times n$ matrix and $\mbox{vec}$ is the vector/stack operator. I have tried using ...
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3answers
26 views

Unit vector c perpendicular [closed]

Find a unit vector $c$ perpendicular to both of the vectors $a = 0j + 1j - k$ and $b = 2i + 2j – k$. Just need steps/hints or even the solution would help me check if I go it right.
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1answer
67 views

An excercise problem on Hoffman Linear Algebra

Let $V$ be the vector space over $\mathbb R$ of all functions $f :\mathbb R \to\mathbb R$, then identify if the following is a subspace of $V$: All $f \in V$ such that $f(x^2)=f(x)^2$ While I ...
1
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1answer
37 views

Construct a linear map $M : V → V$ with the property that $K = \{v ∈ V\mid Mv = 0\}.$

"Suppose that V is a vector space and $L : V → V$ is a linear map. (i) Let K ⊂ V be the set of all vectors $v ∈ V$ such that $L(v) = −v$. Show that K is a subspace of V . (ii) Construct a linear ...
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1answer
31 views

Clarification of ideas concerning a quotient space.

Suppose I have a vector space $V$, and I identify $x\in V$ with $\lambda x\in V$, where $x\neq 0$ and $\lambda>0$, $\lambda\in\mathbb{R}$. I'm confused about two things: (1) Can I define a norm on ...
3
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3answers
160 views

determinant of the linear transformation $T(X) =\frac{1}{2} (AX+XA)$

Let $V$ vector space of all matrices $3\times3$, and let $A$ be the diagonal matrix : $$ \begin{pmatrix} 1 & 0 & 0\\ 0 & 2& 0 \\ 0 & 0& 1\end{pmatrix} $$ Compute thee ...
20
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4answers
14k views

How to find basis for intersection of two vector spaces

What is the general way of finding the basis for intersection of two vector spaces? Suppose I'm given the bases of two vector spaces U and W: $$ \mathrm{Base}(U)= \left\{ \left(1,1,0,-1\right), \left(...
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1answer
20 views

Definition of complex vector space from Rudin RCA

This is definition of complex vector space from Rudin's book. He write that to each pair $(\alpha,x)$, where $x\in V$ and $\alpha$ is scalar there is associated a vector $\alpha x\in V$. That's right. ...
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3answers
59 views

Why a linear trasformation doesn't depend on the bases we choose

Imagine we are given the following linear transformation: $f(x,y) = (x+y, x)$ Imagine we choose a base, let's call it $B_{1}$ and we apply the function to some vector. Now imagine we choose another ...
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0answers
14 views

Show that V1 is a linear subspace of R[x]?

"Let $\mathbb{R}$ be the set of polynomials, and let $ V_1 = (a_1x + a_2x^3 + a_3x^5$ | $a_1, a_2, a_3 \in \mathbb{R}$ ) and $ V_2 = (b_1x^2 + b_2x^3 + b_3x^4$ | $b_1, b_2, b_3 \in \mathbb{R}$ ) be ...
1
vote
2answers
25 views

Show that $V = \ker T \oplus \operatorname{im}T$ where $T$ is an idempotent linear operator [duplicate]

I have to prove that if $T$ is an idempotent ($T^2=T$) linear operator then space $V = \ker T\oplus\operatorname{im}T$. My first try was to think about the basis of subspace $\ker T$. Let say $(e_1,...
9
votes
1answer
163 views

Invariant vectors of $A^n B^m$ with $A,B$ orthogonal matrices

Let $A$ be the following matrix:$$A=\dfrac{1}{2}\ \left( \begin{array}{cccccccccc} -1 & -1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 0 &...
0
votes
1answer
31 views

Non-negative Linear Span of Vectors

I would like to understand if there is a common concept of a `linear span' of a set of vectors which are combined with non-negative multipliers. I know that usual definition of the span of a set of ...
21
votes
2answers
2k views

Cardinality of a Hamel basis of $\ell_1(\mathbb{R})$

What is the cardinality of a Hamel basis of $\ell_1(\mathbb R)$? Is it deducible in ZFC that it is seemingly continuum? Does it follow from this that each Banach space of density $\leqslant 2^{\...
0
votes
4answers
57 views

How to prove a $W=\{(x,y):3x+y=0\}$ is a subspace of $ R^2$

How can I prove this vector $W$ is a subspace of $\mathbb{R}^2$ (closed under addition and scalar multiplication) if I have the condition $3x+y=0$. Does this mean this vector already has the $0$ ...
6
votes
4answers
603 views

Is $\mathbb{C}^2$ isomorphic to $\mathbb{R}^4$?

Are $\mathbb{C}^2$ and $\mathbb{R}^4$ isomorphic to one another? Two vector spaces are isomorphic if and only if there exists a bijection between the two. We can define the linear map $T: \mathbb{C}^2 ...