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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74 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
35 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
53 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
46 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
44 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
48 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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47 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
49 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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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 ...
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3answers
89 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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1answer
25 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
55 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
40 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
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 ...
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3answers
47 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
24 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 ...
1
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1answer
35 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
24 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
19 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
21 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
62 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
27 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.
1
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1answer
38 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
60 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
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1answer
24 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
26 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,...
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1answer
33 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
votes
4answers
62 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 $0$ ...
0
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4answers
50 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
37 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 ...
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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
43 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 ...
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2answers
22 views

Subspace proof wording

not sure how to word my answer for this question: Let V be a vector space and let H and K be two subspaces of V. Show that the following set W is a subspace of V: W={u+v: u ∈ H, v ∈ K} I'm pretty ...
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0answers
30 views

Linear Algebra Axler, 3e, Exercise 1c, P11

P11. Prove that the intersection of every collection of subspaces of V is a subspace of V. My solution was very similar to the one from linearalgebras.com -- Solution: Assume $U_i$ are subspaces of $...
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1answer
36 views

What's the difference between $V\times W$ and $V\otimes W$ where $V$ and $W$ are vector space?

Let $V,W$ vector spaces. I don't really understand what is $V\otimes W$. To me it looks the same that $V\times W$. Do you have any explanation ?
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3answers
42 views

Set of orthogonal vectors in $\mathbb{R}^n$

How can we show that a set of pairwise orthogonal vectors in $\mathbb{R}^n$ has size at most $n$? I know it seems very intuitive, but not sure what the formal proof would look like (whether "...
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0answers
21 views

when can i move a sum through this tensor product?

If I have a vector space $V^{(1)}\otimes V^{(2)}$ and I have some ray $\sum\limits_k x_k s_k\otimes s'_k = s\otimes \sum\limits_k x_k s'_k$, is the only solution that $s_k = s$ $\forall$ $k$? All $x_k$...
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2answers
33 views

vector and curl identity

This popped up in my notes and the author made no remarks about the properties used $\bigtriangledown \times \left ( \vec{E}+\frac{\partial \vec{A}}{\partial t} \right )=\vec{0}$ Then, $\...
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0answers
26 views

Prob. 4, Sec. 4.3 in Kreyszig's functional fnalysis text: Application of the Hahn Banach Theorem

Let $X$ be a real or complex vector space, and let $p \colon X \to \mathbb{R}$ be a real-valued function satisfying $$p(x+y) \leq p(x) + p(y) \ \mbox{ for all } \ x, y \in X$$ and $$p(\alpha x) = \...
3
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1answer
50 views

Dimension of set of $3\times 3$ matrices?

Calculate the dimension of the image and kernel of each linear transformation. (Hint: you do not need to find a matrix representing the linear transformation.) $(a)$ $P\colon\Bbb R^3\to\Bbb R^3$ by (...
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1answer
52 views

Subspace proof $\{\,f \colon\Bbb R \to\Bbb R \mid f(x + 1) = f(x) + 1\,\}$

I have no idea how to show that this is a subspace. Isn't $f(x)=x$ and $f(x)=3x$ a counter-example? It is not closed under scalar multiplication? But I guess it is.. $[e]$ forgot to say that the ...
0
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1answer
51 views

A question about diagonalizable linear operator

Suppose $T$ is a diagonalizable linear operator on a finite dimensional vector space $V$. Prove $V$ is T-cyclic subspace of itself iff every characteristic subspace of it is one-dimensional. It ...
0
votes
1answer
34 views

switching between P3 and R4 vector spaces

first of all I couldn't find a better way of describing what I really meant so here it goes. Lets say I have a vector space $\mathbb{P}_3[\mathbb{R}]$ and a sub vector space $U = \textrm{span}\{ x^2 +...
0
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0answers
21 views

Hermite Polynomials using gram-schimdt

How does one generate these polynomials using the gram-Schmidt algorithm? I know how it should work, but I get 0 as the value for the scalar product of (p1,q0) and q1 should be 2x not x. $$q1\left(x\...