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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14 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 ...
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3answers
39 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 ...
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1answer
19 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 ...
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2answers
88 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 motivation behind word quotient.
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1answer
16 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 $\|\cdot \|$?
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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 ...
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2answers
46 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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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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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 ...
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3answers
140 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....
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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 ...
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1answer
33 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 ...
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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 $(Ax)t = t^2x(a)$, ...
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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
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
45 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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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 ...
1
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2answers
31 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 ...
2
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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
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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 ...
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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 ...
2
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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}(...
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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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=...
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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 \...
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 {$T(u): u \in X$} ...
3
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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 ...
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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, ...
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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 ...
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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
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 ...
2
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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 ...
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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. ...
2
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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 ...
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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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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,...
0
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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 ...
0
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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 $0$ ...
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4answers
48 views

Direct sum of vector subspaces equals $\mathbb R^3$

I tried solving the following linear algebra problem, I hope that someone can tell me if this is a good solution, and if not, how I should solve it. Let $U$ and $W$ be vector subspaces of the vector ...
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0answers
36 views

How to fill the basis of vector subspace up to $\mathbb{R}^3$?

So, if we're given a vector subspace $V$ of $\mathbb{R}^3$ with basis: $$B_V=\{(-1,1,1),(2,1,-1)\}$$ How can we find a basis of vector subspace $F$ such that: $$V\oplus F=\mathbb{R}^3 \ ?$$ What I did ...
1
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1answer
26 views

what is the dimension?

The dimension of the row space of a $8\times 8$ matrix $A$ is 5.if $\mathbb{R}^{8\times 10}$ is a vector space of $8 \times 10$ matrices with real entries. Then $S_{A} = \{ B \in \mathbb{R}^{8\times ...
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0answers
41 views

Action of a Linear Functional on a Polynomial

I was hoping to find a good canonical reference for the mathematics behind something called the action of a linear functional $L$ on a polynomial $p(x)$ which is denoted $\langle L|p(x)\rangle$ ...
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0answers
36 views

What operator has these algebraic properties?

I am working in a space $V$ of objects that behaves like a vector space with a partial ordering $\preceq$. I have discovered an operator $f:V\times V \rightarrow V$ with the following properties: For ...