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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414 views

difference between parallel and orthogonal projection

i would like to understand what is a difference between parallel and orthogonal projection?let us consider following picture we have two non othogonal basis and vector A with ...
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553 views

Prove that the field F is a vector space over itself.

How can I prove that a field F is a vector space over itself? Intuitively, it seems obvious because the definition of a field is nearly the same as that of a vector space, just with scalers instead of ...
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4answers
153 views

Intuition behind definition of transpose map

The linear map $T^t:V' \to V'$ (where $V'$ is the set of all linear maps from $V$ to its scalar field $\Bbb F$) is defined by : $$T^t(f)(v)=f(T(v))$$ This looks like some "commutative" definition ...
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2answers
178 views

Show that the area vectors for a general $n$-sided closed shape sum to zero

It is possible to show that the sum of the area vectors for a general, closed, $n$-sided figure in $\mathbb{R}^3$ (3-space) is zero. Hint: it may be easiest to consider orientable and non-orientable ...
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2answers
127 views

Inconsistent System of Linear Equations

Let $A ∈ M_{n\times n}(F)$. Suppose that the system of linear equations $AX = B$ has more than one solution. Prove that there is a column $C ∈ F^n$ such that the system of linear equations $AX = C$ is ...
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1answer
99 views

angle between steepest gradient of two plane

IF I have two 3d planes such as Oab and Oa'b'. If these two planes intersect a horizontal plane and the intersection of each plane makes AB and A'B' lines. then, Does the angle between AB, A'B' ...
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0answers
174 views

Vector Projection and Cross Product

Let there be $w, u, $ and $v$, such that: $$w \times u = \langle1, 3, 5\rangle$$ $$w \times v = \langle 2, 4, 6\rangle$$ Find: $$v \cdot (((u \times w) \times v) + \text{VP}uv(w)) + ((u ...
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1answer
82 views

Well-definedness of a linear mapping

I have the following theorem: Theorem (from Schaum's Linear Algebra) Let $V$ and $U$ be vector spaces and $\{v_1, \ldots, v_n\}$ be a basis on $V$. Let $\{u_1,\ldots, u_n\}$ be arbitrary vectors ...
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4answers
375 views

Get the direction vector passing through the intersection point of two straight lines

Let say I have this diagram, How to find the direction vector passing through the intersection point of two straight lines? Update: new vector is the bisector of two lines and vector may be ...
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1answer
176 views

Finite dimensional vector space with subspaces [duplicate]

Possible Duplicate: Could intersection of a subspace with its complement be non empty. Is it possible for a finite dimensional vector space to have 2 disjoint subspaces of the same ...
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1answer
166 views

What is an honest basis?

In a comment to this question, the commentator stated that "the monomials form an honest basis for your vector space". To be honest, I never heard of that. Is this something elementary?
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1answer
149 views

Is this argument valid? — bilinear maps

Does this argument make sense? So I have a bilinear form $B:V\times V\to F$. (BTW, is bilinear form equivalent to a bilinear map?) where $V$ is a finite-dimensional vector space. I have a subspace ...
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1answer
691 views

Show that this is a vector space and determine the dimension

Question: Let $v = (1,1,0,1) \in \mathbb{R}^{4}$ Let $$V:= \{f: \mathbb{R}^{4} \rightarrow \mathbb{R}^{2} \mbox{ linear } | f(v) = 0\}.$$ (a) Show that $V$ forms a vectorspace. (b) Determine ...
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1answer
39 views

Same row space is equivalent to same column space?

If $A$ and $B$ are $n \times n$ matrices that have the same row space, then $A$ and $B$ have the same column space. This is false of course. I could just come up with examples though. Can one prove ...
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1answer
47 views

A question about vector fields and divergence

I am reading the paper http://www.goshen.edu/physix/mathphys/gco/TensorGuideAJP.pdf in order to gain a basic understanding about tensors. I had some difficulties about understanding some definitions. ...
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1answer
32 views

Show that $f(x)$ is orthogonal to $f'(x)$ in $L^2(-\pi, \pi)$

I have the following problem: Suppose $f$ is of class $C^{(1)}$, $\;2\pi$-periodic, and real-valued. Show that $f'$is orthogonal to $f$ in $L^2(-\pi, \pi)$ by a) expanding $f$ in ...
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1answer
120 views

Finding the dimension of subspace span(S)

Problem: Consider the set of vectors $S= \{a_1,a_2,a_3,a_4\}$ where $a_1= (6,4,1,-1,2)$ $a_2 = (1,0,2,3,-4)$ $a_3= (1,4,-9,-16,22)$ $a_4= (7,1,0,-1,3)$ Find the dimension of the subspace $span(S)$? ...
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1answer
53 views

How to rotate a 3d vector to be parallel to another 3d vector using quaternions?

I have a vector (a,b,c) and another vector (d,e,f). I'm trying to rotate (a,b,c) so its parallel to (d,e,f) using quaternions. I need help understanding how I would do this. I have so far that a ...
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0answers
30 views

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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1answer
23 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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1answer
292 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
65 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
381 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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2answers
146 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
96 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
145 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
1k 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
164 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
122 views

$\operatorname{rank}(A + B) ≤ \operatorname{rank}(A) + \operatorname{rank}(B)$ [duplicate]

Let $A, B ∈ M_{m×n}(F)$. Could someone give a hint as to how to prove that $$\operatorname{rank}(A + B) ≤ \operatorname{rank}(A) + \operatorname{rank}(B).$$
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1answer
41 views

Find normal to plane in a point

Let's suppose I have a point $P(x,y,z)$ that I know for sure it lies on a plane $$n_x(x-x_0) + n_y(y-y_0) + n_z(z-z_0) = 0$$ How can I calculate the plane's normal vector in that point?
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462 views

extending a linearly independent set to a basis

I want to show that every linearly independent set in a finite-dimensional linear space can be extended to a basis for the entire space.
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2answers
82 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
183 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
152 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
85 views

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
129 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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3answers
2k views

Equation of a plane equidistant from two points $A$ and $B$?

Suppose I have two points $A=(x_1,y_1,z_1)$ and $B=(x_2,y_2,z_2)$. Suppose I want to find the equation of the plane equidistant from these points of form $ax+by+cz+d=0$. What is the equation in terms ...
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2answers
969 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
209 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
702 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
80 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
97 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
271 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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2answers
231 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 ...