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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matrices forms a basis for vector space 2x2

$\begin{bmatrix}0&1\\2&3\end{bmatrix}$ $\begin{bmatrix}3&4\\5&6\end{bmatrix}$ $\begin{bmatrix}7&8\\9&10\end{bmatrix}$ $\begin{bmatrix}11&12\\13&14\end{bmatrix}$ Show ...
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0answers
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Understanding 2nd half rank-nullity theorem proof.

I'm trying to understand the second half of the rank-nullity theorem (the part that shows $T(e_{k+1}) \dots T(e_{k+r})$ is independent). Assume $e_1 ,\dots e_k, e_{k+1}, \dots e_{k+r}$,is a basis for ...
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22 views

Partly Orthogonal of a basis

I have basis set ${\cal K}=\{v_1,\dots,v_{k-d},b_1,\dots,b_d\}$ of $\mathbb{R}^k$. I'd like to get $\cal W=\{w_1,\dots,w_d\}$ by linear combinations of the elements of $\cal K$ such that $\cal K\cup ...
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377 views

What does this linear algebra notation mean?

I'm trying to prove that a particular $V$ is a $\Bbb{Q}$-vector space. The question says to take the element $0_V = 1$, the function $+_V : V \times V \to V$ given by the function $[x +_V y = xy]$, ...
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277 views

Proof: dimension of annihilator

First there is a vector space V and U is vector subspace of V. Furtermore $U^{0}$ is the annihilator of U (= {$\varphi \in V^{*} |\space\forall u \in U: \varphi(u) = 0$}). I need to show that: dim(V) ...
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1answer
58 views

Eigenspace and polynomials?

My prof introduced us to eigenvectors and eigenvalues today. He then gave us the following theorem: Theorem 6.6: Let $A$ be a square matrix, let $\gamma$ be an eigenvalue of $A$ with multiplicity ...
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1answer
365 views

Dimension of direct sum of vector spaces [duplicate]

Let $V$ and $W$ be finite dimensional vector spaces on a field $F$. Show that $\dim(V\oplus W) = \dim V +\dim W$. My idea: let $\dim V=n$ and $\dim W=n$. So $\mathcal{A}=$ {${v_1 , v_2 ,... , ...
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141 views

A subspace contains the zero vector; intersection of subspaces is a subspace

I have a simple question I have to answer but I am not sure where to start with this due to my lack of experience regarding subspaces. Can anybody help me? Assume $V \subset \Bbb R^n$ is a ...
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2answers
95 views

Is there a linear transformation who domain isn't all of $\mathbb{R}^n$?

My prof said that for a linear transformation: $$T: \mathbb{R}^n \rightarrow \mathbb{R}^m$$ for some real $n$ and $m$, $\mathbb{R}^n$ is called the domain. But some "normal" functions have domains ...
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1answer
52 views

I need to find whether it is one and onto.

$X=C[0,1]$ define $T:X\to X$ by $T(f(x))=\int_{0}^{x} f(t) dt$ Then I need to find whether it is one and onto. If $T(f(x))=0$ then $\int_{0}^{x} f(t) dt=0$ taking derivative we get $f(x)=0$ so ...
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1answer
141 views

What is $\mathrm{Sym}(V)$, where $V$ is a vector space

What is $\mathrm{Sym}^n(V)$, where $V$ is a vector space? (I've found a problem list and have some problems on notations)
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1answer
1k views

Finding an orthonormal basis using Gram Schmidt process

OK, here's a question with polynomials. We want to find an orthonormal basis using Gram Schmift. Assuming that we are in a vector space V, $R^2[X]$ where {$f = \lambda_0+\lambda_1X+\lambda_2X^2$}. ...
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2answers
69 views

Computing cross product using norm and angle

Sorry for the weird title, if someone finds a better title for my problem be my guest to edit it ;) For $\mathbf{v,w} $ in R³ with $\mathbf{||v||=1 ;||w||=4; \theta =\frac{2\pi}{3}}$ Solve ...
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3answers
909 views

Vectors that form a triangle!

I have a problem here. How can I prove that sum of vectors that form a triangle is equal to 0 $(\vec {AB}+\vec {BC}+\vec {CA}=\vec 0)$ ? Thank you!
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2answers
119 views

Is $w$ a vector space?

Let $W$ be the set of all solutions$(X,y,i,r)$ such that $a+b=m^2$. is $w$ a vector space? Can anybody do the whole thing for me and explain shortly every step? I have to do this kind of lots of ...
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1answer
158 views

least square solution for obtaining a line 3d by intersecting many planes

If I have been given 4 planes and I know only a point (just a point lie on the plane e.g. o1,o2, o3 & o4) and normal vector of each plane (n1, n2, n3 & n4). Then, by intersecting all 4 (or ...
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2answers
81 views

Finite Dimensional Subspaces and Their Properties

Let $W_1$ and $W_2$ be finite dimensional subspace fo a vector space $V$. How should I start to prove that the subspaces $W_1 \cap W_2$ and $W_1+W_2$ are also finite dimensional and \begin{eqnarray} ...
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2answers
181 views

what is $ M^{\perp}$ given set?

Let $ ‎X=C[-1,1]‎$‎‎ be inner product space with definition $$‎\langle f,g‎‎‎\rangle =‎\int_{-1}^1 f‎‎ \overline{g}‎ ‎dt ‎‎.$$ Let $M$ be the subspace defined by ‎$$ ‎M= ‎‎\left\{f‎ \in ‎X\mid ...
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2answers
136 views

Set of points reachable by the tip of a swinging sticks kinetic energy structure

This is an interesting problem that I thought of myself but I'm racking my brain on it. I recently saw this kinetic energy knick knack in a scene in Iron Man 2: ...
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1answer
57 views

New values of vector after change of base

We make a change of base with the matrix $$S=\left[\begin{matrix}p & q \\ 1 & 1\end{matrix}\right]$$ so the vector $x= (8, 3)$ of $\mathbb{R}^{2\times 1}$ becomes $x= (1, 2)$ and the vector ...
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0answers
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Finding a Vector not in a Collection of Proper Subspaces

Let's take $K$ to be an infinite field and $V$ a vector space over it. Allowing $U_1,...U_l$ to be $l$ proper subspaces of $V$ (none the same), and further assuming that for all $i$, $U_i$ is not in ...
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1answer
122 views

Orthonormal basis with three vectors

So I have the following question: $Let \\ u_1 = \begin{pmatrix} \frac{1}{3\sqrt{2}}\\\frac{1}{3\sqrt{2}}\\\frac{-4}{3\sqrt{2}}\end{pmatrix} u_2 = \begin{pmatrix} ...
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2answers
947 views

How to prove that the union of two subspaces must be subsets of each other? [duplicate]

Possible Duplicate: Union of two vector subspaces not a subspace? $U,W\subseteq V$ are subspaces. Prove that in order for $U \cup W$ to be a subspace as well, either $U\subseteq W$ ...
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2answers
204 views

Vector Analysis & Linear Algebra

I'm given a positive number, a unit vector $u \in \mathbb{R} ^n $ and a sequence of vectors $ \{ b_k \} _{ k \geq 1} $ such that $|b_k - ku| \leq d $ for every $ k=1,2,...$. This obviously implies $ ...
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2answers
683 views

Can a Vector space have subspaces of same dimension over different fields?

Just wondering if a finite dimensional vector space could have two subspaces such that each of these subspaces has the same dimension but form vector spaces over different fields ?
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1answer
79 views

Help needed with tensors [duplicate]

Possible Duplicate: An Introduction to Tensors Recently I came across the concept of tensors and heard it is very difficult to understand. Is there a ...
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1answer
92 views

How to draw arrows by rotating lines in 3d space?

I am trying to figure out direction vectors of the arrowheads of an arrow. Basically I'm given a normalized direction vector ...
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1answer
248 views

Necessary condition of a vector space having only one basis? [closed]

I want to know when a vector space has only one basis.
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226 views

$\mathbb{C}^3$: Orthogonal Complement

Let $S=\{(1,0,i),(1,2,1)\}$ in $\mathbb{C}^3$. What is the method used to find a basis for $S^{\perp}$? EDIT$^1$: I think this bit of literature from Gockenbach's Finite-Dimensional Linear Algebra ...