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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determine where a vector will intersect a plane

I have a vector with position $O=(o_1,o_2,o_3)$ and direction $D=(d_1,d_2,d_3)$ and a plane determined by 3 points $A=(a_1,a_2,a_3),B=(b_1,b_2,b_3),C=(c_1,c_2,c_3)$. In which point will the vector ...
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2answers
58 views

matrix to vector

What's the formal way to map a Matrix $A \in M(n \times n, K)$ to a row vector $B \in K^{n²}$ where a) the columns $col_i(A)\quad, \quad 1 \leq i \leq n$ are arranged one below the other ...
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2answers
962 views

Finding an orthonormal basis in a subspace

I have a problem that says: "By using the Gram Schmidt process(if you need it), find an orthonormal basis B in the subspace: $A=\{ u \in R^3 \ | \ x_1-4x_2-x_3=0 \}$ I chose as the basis, $B=\{[1 \ ...
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1answer
164 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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100 views

Is a multilinear form/mapping a product of some type on vectors?

Added: are all types of mappings for vector spaces with "product" in their names always multilinear mappings between some vector spaces? Are there many counterexamples? $F$ is a field. Any ...
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3answers
168 views

What do I call a unit vector parallel to a coordinate axis?

What do I call an arbitrary element of this set of vectors? $$ \begin{align*} \{&\langle 1, 0, 0 \rangle, \\ &\langle 0, 1, 0 \rangle, \\ &\langle 0, 0, 1 \rangle, \\ &\langle ...
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1answer
1k views

Finding a basis for the intersection of two subspaces

I'll try to write this as best as I can... Let the following $U_1, U_2$ be subspaces of $\mathbb{R}^4$ $$ U_1 = \begin{Bmatrix} (x, y, z, w) : z-y+2w = 0 \end{Bmatrix} $$ $$ U_2 = ...
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0answers
106 views

How to a find vector which statisfies $L^1$ and $L^2$ norm?

Is there any algorithm, when given a vector $x$ of $n$ dimension, $k_1$($L^1$-norm) and $k_2$($L^2$-norm), find the vector $v$ of dimension $n$ having $\sum_i(|v_i|)=k_1$ and $\sum(v_i^2)=k_2$ which ...
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0answers
324 views

3D Mathematics Projecting One Vector onto Another

I have a Vector function which takes two Vectors and and attempts to "project" these vectors together. The method used is to separate the larger of the two vectors (the larger value calculated via ...
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2answers
171 views

What's a good method to solve for scalars in a vector equality?

For example, what's a good way to solve for $c_1$, $c_2$ and $c_3$ in: $$ c_1(1,-1,0) + c_2(3,2,1) + c_3(0,1,4) = (-1,1,19) $$
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0answers
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Self-absorbing subsets in a vector space

From planetmath Let $V$ be a vector space over a field $F$ equipped with a non-discrete valuation $|\cdot|:F\to \mathbb{R}$ . Let $A$ and $B$ be two subsets of $V$. Then $A$ is said to absorb ...
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2answers
728 views

Scalar Product for Vector Space of Monomial Symmetric Functions

Suppose a multinomial $P(X_1, X_2,\ldots, X_n)$, that is given as a sum of monomials $m_\lambda$ with coefficients $c_k$: $$ P(\vec{X})=P(X_1, X_2,\ldots, X_n) = \sum_k c_k m_{\lambda_k} . $$ Since ...
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4answers
10k views

The shortest distance between two parallel lines

I was working on a set of problems involving finding the shortest distance between two skew lines, which was fine, but then parallel lines showed up. In essence this should be much easier to solve ...
5
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4answers
122 views

Vector dimension of a set of functions

Let $F$ be a field and $S$ an infinite set. Set $V=\{f:S \rightarrow F\}$ endowed with the vector space structure that results from the pointwise operations of $F$. It is easy to prove that $|S| \leq ...
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1answer
46 views

Asymptotic convergence

I have a previous similar question. I'm working out that one with the answerer, but I'm trying to gain insight from a different angle, especially in approaching these problems. I must establish that ...
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1answer
85 views

Linear Algebra: Minimum Polynomials for Operators on $\mathbb{R}[x,y]$.

I've been working through some problems set by my University over the past few years, and have encountered this problem. Problem Let $n > 1$ and let $V_n$ be the subspace of $\mathbb{R}[x, y]$ ...
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3answers
402 views

The dimensions of $V$ and $V^\perp$ are complementary

Statement: Let $V$ a vector subspace of $\mathbb{R}^4$. The bilinear form $f(x,y) = x_1y_1 + x_2y_2 + x_3y_3 - x_4y_4$ where $x,y \in \mathbb{R}$. Let $V^\bot = \{ y \in \mathbb{R}^4 : f(x, y) = 0 \ ...
3
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2answers
109 views

Why is the maximal value attained at the boundary?

Let $A$ be a real matrix. Denote $\|\cdot \|$ the $p=1$ norm (sum of absolutes of the elements). Let $C$ be all vectors (of compatible size with $A$) whose elements are in the range $[-1,1]$ How to ...
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1answer
350 views

Geometry problem from a Berkeley course

I've been trying to solve this problem proposed as part of one of the first lectures of a Berkeley linear algebra course: "What Good is a Basis ? The freedom to choose a basis often simplifies ...
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1answer
442 views

When are two diagonal matrices congruent?

This is probably a question that does not admit a simple answer. However, I'd like to know whether there exist criteria that determine when two diagonal matrices are congruent. I have the suspicion ...
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3answers
358 views

Algebra: Orthogonal Complement

Problem Let $V$ be a real inner product space and $U \subset V$. Show that $(U^{\perp})^{\perp}=U$. Progress Clearly for $x\in U$ we have that $\langle x,v \rangle=0$ for all $v \in ...
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2answers
938 views

Is the cross-product of two displacement vectors orthogonal to both of them?

I have the point P,Q and R given. I calculate the displacement vectors PQ and PR. If I then compute their cross product I get a vector orthogonal to the plane they're in. But the value of the cross ...
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1answer
269 views

Calculating two perpendicular vectors between two points (maybe vector decomposition?)

I think this is a fairly basic question, but I'm pretty bad at math so I've spent several hours trying to work this out using my poor trig skills since I don't understand vector math at all. Maybe ...
4
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1answer
14k views

Proof for triangle inequality for vectors

Generally,the length of the sum of two vectors is not equal to the sum of their lengths. To see this consider the vectors $u$ and $v$ as shown below. By considering $u$ and $v$ as two sides of a ...
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1answer
630 views

Linear Algebra: Dual Basis Problem

Problem Let $V$ be the vector space of all polynomial functions $p$ from $\mathbb{R}$ to $\mathbb{R}$ which have degree two or less. Define three linear functionals on $V$ by ...
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1answer
107 views

Vector Spaces Question

A) Is it true that always the group $\{(Z_1,Z_2) \in \mathbb{C}^2\;|\;Z_2 \in \mathbb{R}\}$ is a subspace of $\mathbb{C}^2$ as a vector space over $\mathbb{C}$? B) Is it true that always the group ...
2
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0answers
98 views

Condition on $c$ for a contraction map

I am extending Example 2.2 on this sheet. Suppose $f(x(s),s)$ is such that $|f(x(s),s)-f(y(s),s)|\leq K |x-y|$ for some $K>0  ---(1)$ and $x,y\in C[0,t_f]:\,\,\,t_f<\infty$ Also, let $T$ ...
3
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2answers
191 views

Nonsingular bilinear map

This is related to a previous question I asked. But I realize that my logic there is total bonkers. And it would be great if someone could help me out a bit. $B:V\times V\to F$ is a bilinear form ...
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1answer
219 views

Combining “real world” coordinates, and “personal” acceleration to get “real world” velocity

I'm working on a mobile phone app idea the moment. This app is currently being prototype on a Windows Phone 7 device. The Motion API on that device gives me frequent updates on : the current 3 ...
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5answers
208 views

Why is there implied an equality between vectors and $n$-tuples?

Are they considered equal in some sense? For instance, I always write "...for vector $\mathbf{x} \in {\mathbb{R}}^n$ we have ...". I have a small problem with this (not a big one). The problem comes ...
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3answers
2k views

How do you find the distance from a point to a plane?

I am having trouble with this: Find the distance from the point $(1,1,1)$ to the plane $2x+2y+z=0$. Any ideas? Thanks.
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1answer
91 views

Bases and inner products

I am not quite sure what this question is asking for: Given $f(\vec{x})=x^2+xy+y^2+yz+z^2+xz$, find a basis for the corresponding inner product on $\mathbb R^3$. (I was told that there is an ...
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1answer
146 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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0answers
72 views

clever solution to decomposition of linear products?

There may be a better name for this class of problem, and if so feel free to edit! Imagine a matrix consisting of the following columns: daily return, $\alpha_t$, $factor^1_t$, $factor^2_t$, ... and ...
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1answer
633 views

Annihilators and dimensions

This question is related to this question. I have learnt from the question in the link that the equality below does not always hold. So here is my new question: What is a necessary and sufficient ...
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1answer
165 views

Dimensions of vector subspaces

Given a bilinear map $B:X\times Y\to F$ where $X,Y$ are vector spaces and given $S\leq X$, why is $\dim S+\dim \operatorname{ann}(S)=\dim Y$ where $\operatorname{ann}(S)$ is the annihilator of $S$ ...
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1answer
141 views

Find all endmorphisms $f$ of a real vector $V$ space such that $f\circ f=\operatorname{id}_V$

Find all endmorphisms $f$ of a real vector $V$ space such that $f\circ f=\operatorname{id}_V$ The problem is trivial if $V$ is a $\mathbb{C}$-vector space. Yet here $V$ is a vector space over ...
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1answer
91 views

What is the proper term for the entity that relates a vector space and a set?

One way to generate a metric for a set $S$ (a distance function between elements $a,b$ of the set $S$) would be by associating it with a vector space $V$ (the vectors that connect the elements $a,b$) ...
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1answer
2k views

Vector Mid Point vs Mid Point Formula

Given $OA=(2,9,-6)$ and $OB=(6,-3,-6)$. If $D$ is the midpoint, isit $OD=((2+6)/2, (9-3)/2, (-6-6)/2)$? The correct answer is $OD=\frac{1}{2}AB=(2,-6,0)$
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1answer
1k views

Normal Form Vector Equation for Plane Containing $A(-2,1,5)$ and $\text{Line: } (2-\mu)i+(3+\mu)j+(4+\mu)k$

I am doing part (ii) of Find normal form equation of plane containing $A(-2,1,5)$ and $\text{Line: } (2-\mu)i+(3+\mu)j+(4+\mu)k$ I did: $AB=4i+2j-k$ $\text{normal} = ...
2
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2answers
301 views

Basis for the intersection of subspaces

Prove or disprove the following; Let $V$ be a vector space and $U$ and $W$ two subspaces of $V$. If the set of vectors $\{b_1,\ldots b_n\}$ is a basis for $U\cap W$, then it is also a basis for both ...
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2answers
27 views

Definition clarifications on functions of vectors and their derivatives

Definition clarifications would be appreciated: How do I interpret the following ? For $f: R^n\to R^m$, $Df(\vec\xi)(\vec{x})$ in differentiation of a vector function. I know it as a function that ...
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2answers
102 views

Multiplying Complex Numbers by i

But I am wondering why isit $PQ \perp QR$ and not $QP \perp QR$ as shown below? UPDATE How do I get the equation: $(i-1)b=ic-a=i(1-2i)-(-1+4i)=3-3i$? Where does $(i-1)$ come from? I dont ...
2
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2answers
303 views

Definition clarification about dimension of a subset

Does it make any sense to talk about the dimension of a subset (not necessarily a subspace) of a vector space at all? What if say you can pick $n$ linearly independent vector from the subset but the ...
5
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3answers
293 views

How to think of a function as a vector?

In order to apply the ideas of vector spaces to functions, the text I have (Wavelets for Computer Graphics: Theory and Applications by Stollnitz, DeRose and Salesin) conveniently says Since ...
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2answers
2k views

Finding a point along a line in three dimensions, given two points

I need to find a point along a line segment in three-dimensional space, given two points. For example: Find a point along a line segment between point $a(-2, -2, -2)$ and $b(3, 3, 3)$ which is at ...
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1answer
65 views

A question about pyramids (polytopes)

Let's use the following definition of a face: A nonempty convex subset $F$ of a convex set $C$ is called a face of $C$ if $\alpha x + (1-\alpha) y \in F$ with $x, y \in C$ and $0 < \alpha < 1$ ...
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2answers
412 views

Two definitions of a face of a convex set: are they equivalent?

I am used to the following definition of a (proper) face of a polytope: A nonempty convex subset $F$ of a polytope $C$ is called a face of $C$ if $\alpha x + (1-\alpha) y \in F$ with $x, y \in C$ and ...
2
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0answers
103 views

Is there a name for the element-by-element multiplication of two vectors? [duplicate]

Possible Duplicate: Is this Vector operation defined? Does it have a name? I am a software engineer, so bear with me. Suppose I have two vectors of equal length containing numbers. Is ...
3
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3answers
16k views

Find the equation of the plane passing through a point and a vector orthogonal

I have come across this question that I need a tip for. Find the equation (general form) of the plane passing through the point $P(3,1,6)$ that is orthogonal to the vector $v=(1,7,-2)$. I would ...