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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1answer
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Calculate the unit normal vectors to the both sides of a plane

Calculate the unit normal vectors to the both sides of a plane passing through three points with coordinates (1,0,1), (1,1,-1) and(-1,1,1). My answer is [$\sqrt{6}/6 , \sqrt{6}/3 , \sqrt{6}/6$] and ...
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3answers
197 views

Scalar Equation of a Plane

Determine the value of $k$ such that the line with parametric equations $x = 2 + 3t, y = -2 + 5t, z = k$ is parallel to the plane with equation $4x + 3y – 3z -12 = 0.$
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1answer
46 views

inner product space , dual space, proof about isomorphism

Let $V$ be a vector space (not necessary being finite dimensional) and let $U,W$ be subspaces of $V$ such that $V = U\oplus W$. Prove that $V^\ast/(W^0)$ is isomorphic to $W^\ast$. Notation and ...
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2answers
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Determine if all vectors of the form (a,b,c), where b=a+c+1 are subspaces of R^3?

Determine if all vectors of the form $(a,b,c)$, where $b=a+c+1$ are subspaces of $\mathbb{R}^3$? Use the theorem: If $W$ is a set of one or more vectors from a vector space $V$, then $W$ is a ...
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0answers
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Get Attitude from 2-axis vector

I've built a quadrotor but my 3-axis accelerometer has a fault, the Z-Axis doesn't work. I would normally get my attitude with the following code pitch = atan2(accel_X, accel_Z)*RadToDeg; roll= ...
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1answer
13 views

equality of annihilators of a vector space

I have a problem as follows: W1 and W2 are subspaces of a finite-dimensional vector space V. W0 is the annihilator of W. (a) Prove W01=W02 implies W1=W2
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1answer
30 views

V = U⊕W then Prove that (V/W)* is isomorphic to W^0

Let $V$ be a vector space (not necessary being finite dimensional) and let $U$, $W$ be subspaces of $V$ such that $V = U\oplus W$. Prove that $(V/W)^*$ is isomorphic to $W^0$. note: (V/W)* is the ...
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1answer
22 views

Why should we expect the divergence operator to be invariant under transformations?

A lot of the time with vector calculus identities, something that seems magical at first ends up having a nice and unique proof. For the divergence operator, one can prove that it's invariant under a ...
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1answer
22 views

Prove Linear Dependence in T: V -> W

Problem: "Let $V$ and $W$ be vector spaces and let $T:V \rightarrow W$ be a linear transformation. Prove that, if $\{v_1, v_2, v_3\}$ is a set of three linearly dependent vectors in $V$, then the set ...
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0answers
25 views

Why is $\hat{x}$ in the linear regression equation $A^TA\hat{x} = A^Tb$ part of $C(A^T)$

When finding the best fit line for a number of points, we use $A^TA\hat{x} = A^Tb$ where we solve for $\hat{x}$. I understand that the projection $p=A\hat{x}$ is part of the column-space of $A$ and ...
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1answer
25 views

Why we take transpose of Vector (Displacement Vector)?

I'm trying to understand some equations that involves transpose of vectors (displacement vectors to be precise) Two set of vectors F and G (with i,j) that corresponds to X,Y value in plane and ...
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1answer
93 views

Problem with alternate solution — Equation of plane through point and containing intersection line of two planes [Stewart P $803, 12.5.37$]

$37.$ Find an equation of the plane that passes through the point $(1, -2, 1)$ and contains the line of intersection of the planes $x + y - z = 2$ and $2x - y + 3z = 1$. $\bbox[3px,border:2px solid ...
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0answers
97 views

Infer distance from a point to a line, from the distance from a point to a plane [Stewart P793 12.4.44]

I'm able to prove $44$, but how would one deduce $43$ from it without further industry, forthwith? $43$ seems like a reduced, 2D version of $44$? I'm not enquiring about individual proofs. $44.$ ...
2
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1answer
19 views

How to get perpendicular vector close to another vector?

Suppose I have two vectors $\vec v_1$ and $\vec v_2$ in $E^3$ space. How can I find a vector $v_3$ such that $\vec v_3$ is perpendicular to $\vec v_1$ the angle between $v_2$ and $v_3$ is minimized ...
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3answers
67 views

An infinite generating set of a finite dimensional vector space contains a basis

Let $S$ be an infinite generating set of a finite dimensional vector space , then how do we prove that there is a subset of $S$ which is a basis of the vector space ? Please help
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1answer
43 views

Question on sequence space (as a linear space)

Let $X$ be the space $\ell_\infty$ of all bounded sequences of real scalars. If $Y$ is the set of all $x\in X$ that have bounded partial sums (1) Can I say $Y$ is a linear space (as a subspace of ...
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2answers
17 views

Finding an Orthonormal Basis using Gram Schmidt

Given the set of vectors $S=${${V_1=\binom{1}{4},V_2=\binom{4}{-4} }$} I am to find an orthonormal basis for $R^2$ using the Gram-Schmidt process. I've already worked it out and found the orthonormal ...
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1answer
233 views

Finding a vector in the plane of one vector and orthogonal to another

I am trying to find vector u, such that it is coplanar with vector a, orthogonal to vector b and displaced from a along vector p. Assume all vectors are unit vectors I can form 3 equations: u.b = 0 ...
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2answers
99 views

What exactly is a linear space?

I often stumble upon books using the term "linear space" (outside of Linear Algebra) and I have never been totally comfortable with this. Perhaps I am over complicating this, but my intuition says ...
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0answers
8 views

How to find optimal perpendicular axis of rotation vector?

I am drawing lines on the screen. Each line has a point (x,y,z) and a direction (u,v,w). I want to draw arrow heads on these ...
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2answers
92 views

Convex cone question.

Hoi, let $V$ be finite dimensional real vector space with inner product $\left\langle . \right\rangle$ and let $\Gamma \neq \{0\}$ be a closed convex cone. Let $$\Gamma_0^{\perp}:=\{v\in ...
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1answer
38 views

How to rotate a line 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
40 views

Farey Sequence Vector Orthogonality Relation Question

Take the Farey sequence $\mathcal{F}_n$ for $n=39$ with values $a_m\in \mathcal{F}_n$ and put them into a vector $$ \vec v_k=\biggr(\exp(2\pi i k a_m)\biggr)_m $$ Since Merten's function for $n=39$ ...
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1answer
17 views

The difference between norm and modulus

I'd like to know the difference between norm of a vector, ||v|| and the modulus of a vector, |v|
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3answers
29 views

A question on basis of vectorspaces and subspaces

Let $V$ be a finite dimensional vector space and $W$ be any subspace . It is known that if $A$ is any basis of $W$ then by "extension-theorem" , there is a basis $A'$ of $V$ such that $A \subseteq ...
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1answer
19 views

Defining operations for a vector space

I was hoping someone could help me with the following. Is it possible to define operations + and $\cdot$ on this set to make it a vector space: \begin{equation*} ...
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0answers
3 views

Gentle introduction to discrete vector field [on hold]

I am looking for a gentle introduction to discrete vector field. Thanks in advance.
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2answers
41 views

Arbitrary (i.e. not necessarily finite-dimensional) vector spaces; reference request.

Its virtually impossible to complete an undergraduate degree these days without studying finite-dimensional vector spaces in quite some detail. So like most of us, I've done all that; however, just ...
4
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4answers
246 views

The Linear Combinations of Two Vectors Fill the Plane Unless _ [Strang P10 1.1.30]

This question is from the 1st chapter of Intro to Linear Algebra, 4th Ed, by Gilbert Strang. So please omit concepts which succeed this question: matrices, rank, REF, nullspace, $Ax=b$, linear ...
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1answer
193 views

Dual operator relationship with complex conjugate.

Let $V$ be a $n$ dimensional vector space spanned by $\{e_{i}\}_{i=1}^{n}$. Let $T:V\to V$ be a linear operator with matrix transformation $A$. Is there any relationship between the dual operator ...
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1answer
630 views

Orthogonal Projection of a Point onto a Plane

I'm dealing with an exercise that requires I find the orthogonal projection of a given point onto a given plane. I don't want an answer directly for my exercise, I would instead like to understand ...
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1answer
34 views

Determining and enforcing linear dependence

Assuming we have a large set of multi-dimensional vectors (20k vectors, 100 dimensions each). My questions are the following: How can we determine the level of linear dependence of this set? Is ...
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0answers
11 views

Transformation Matrix of a function

I have the following: (Note: $V^{*}$ is defined as: $V^{*} = \{ L: V \rightarrow \mathbb{R} | \text{L is linear} \}$) Let $V$ be an $\mathbb{R}$-Vectorspace. Let $\phi \in V^{*} \text{ \ } \{0 \}$ ...
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1answer
26 views

Vectors and Planes

Let there be 2 planes: $x-y+z=2, 2x-y-z=1$ Find the equation of the line of the intersection of the two planes, as well as that of another plane which goes through that line. Attempt to solve: the ...
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0answers
5 views

Geometry of Vectors question

Let there be 3 points; $A(1,2,0), B(2,2,-1), C(4,0,1)$. Find the plane in which all $3$ lie, and find point $D$ such that $ABCD$ is a parallelogram . I did find the plane equation which is ...
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1answer
32 views

What is the canonical basis of a dualspace in $\mathbb{R}^3$?

I have the following: Consider the basis $$B := \{\begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}, \begin{pmatrix} -1 \\ 1 \\ 2 \end{pmatrix}, \begin{pmatrix} 2 \\ 2 \\ 1 \end{pmatrix} \}$$ of the ...
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1answer
19 views

Find a basis for $B/\mathfrak{B}^e$

In the context of algebraic integers, I would like to solve te following problem. Let $A \subset B$ be two rings, $\mathfrak{p}$ a prime ideal of $A$ and $\mathfrak{B}$ a prime ideal of $B$ lying ...
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1answer
18 views

Expected length of projection of vector

How can be the expected length of projection of vector of length $l$ is $l/\sqrt d$ where d is dimension of underlying vector space. I am using $l_2^2$ norm
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0answers
25 views

Vector Question Help

A plane is determined by $(x,y,z) = (1,-1,0) + t(1,-1,2)$ and point $p(1,2,3)$. find point of intersection of $(1,4,-1)+s(-6,2,-4)$ with this plane. I tried this: given the data plane equation is:$$ ...
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2answers
13 views

Determining line equation

Find the equation of the line going through the point $(2,-3,4)$ ,and which is perpendicular to the plane $ x+2y + 2z = 13$ So I tried this: the normal of the plane is $(1,2,2)$, random point on the ...
0
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1answer
55 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
34 views

How to find a transition matrix?

let $a= \{a_1, a_2, a_3, a_4\}$ and $b=\{b_1,......,b_4\}$ and $r = \{r_1,...,r_4\}$ Also, $b_1 = 4a_1$ $b_2 = 8a_1 + 7a_2$ $b_3 = 4a_1 + 4a_2 + 4a_3$ $b_4 = 9a_1 + 5a_2 + 8a_3 + 5a_4$ and ...
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1answer
21 views

Question about vector equations of lines and planes

Find the equation of the line going through the point $(2,-3,4)$ ,and which is perpendicular to the plane $ x+2y + 2z = 13$ So I tried this: the normal of the plane is $(1,2,2)$, random point on the ...
0
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0answers
40 views

Calculating the null space of a matrix [closed]

I am sorry for maybe this is a duplicate question but I really need someone to help me with this I am trying to calculate the null space of this matrix, but I really don't know how and I also have ...
0
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2answers
35 views

Vectors Geometry question

Find the equation of the line going through the point $(2,-3,4)$ ,and which is parralel to the plane $ x+2y + 2z = 13$ So I tried this: the normal of the plane is $(1,2,2)$, random point on the line ...
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1answer
34 views

Uinviersal property of basis of a vector space

Let V be a vector space over a field k. Let B be a subset of V. If any set map from B to any vector space W can be extended uniquely to a k-linear map from V to W. Then B is a basis of V. Can ...
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2answers
43 views

Vectors Question

I have a question regarding Vectors; Find the equation of the plane perpendicular to the vector $\vec{n}\space=(2,3,6)$ and which goes through the point $ A(1,5,3)$. (A cartesian and parametric ...
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0answers
30 views

Question about Hahn-Banach separation Theorem

So here is my question, I am just reading about the Hahn-Banach separtion Theorem and there is one case where a question appeared, namely, Let $X$ be a normed $\mathbb R$ vectorspace and let $A,B$ ...
4
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1answer
85 views

An Algebraic Version of vector spaces

Consider the following set of real numbers $\mathcal{X}=\{1,2,3,\sqrt{2}+1,\pi+\sqrt{2}\}$. Lets consider the set of all linear combinations with integer coefficients of these numbers which I will ...
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1answer
20 views

Show the Hermitian matrices, with trace(g*g1,1)=0 form a vector space.

This is a question from an example sheet that I think may have a mistake in it. Show that the set of Hermitian matrices $A \in H_2 (\mathbb{C})$ with Trace$(A\cdot A_{(1,1)})=0$ is a real three ...