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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Reverse of rotate a vector around an axis - Finding North with magnometer

I've been reading the wikipedia page on rotation matrices and I know the reverse (or at least a very related) version of this question has been asked many times before on this site. However my ...
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1answer
49 views

Codimensionality: On Cardinality of Linear Equations

How does the codimension of a subspace give the number of linear equations needed to define the subspace?
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1answer
46 views

number of elements in vector space

Given that $k$ is a finite field with $q$ elements and $V$ is a $n$-dimensional $k$-vector space, then by basis representation, we know that for $v \in V, v=a_1v_1+a_2v_2+\cdots+a_nv_n$ uniquely. ...
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Magnitude of Axis vectors in question

I have a question on my revision sheet: Write the vector, v=-2 i + 4 j , in polar form. is it safe to assume axis vectors i and j have a magnitude of 1?
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2answers
110 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
65 views

Scalar product with ON-base $e_1,e_2,e_3$

Get the vector u which length is 4, in the ON-base $$e_1,e_2,e_3$$ and the baseangles $$\frac{\pi}{3}, \frac{5\pi}{6}, \frac{\pi}{2}$$
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2answers
69 views

Function space of a finite set and $\Bbb R^n$

I read in a tutorial that a function space $F(S, \mathbb{R})$ of a finite set $S$ of cardinality $n$ has dimension $n$. To be clear $F(S, \mathbb{R})$ is the set of all functions defined on the set ...
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2answers
139 views

$\operatorname{span}(S) = V$, finite dimensional. Does there exist a subset of $S$ which is a basis for $V$?

Let $V$ be a finite dimensional vector space and $S \subset V$ a subset (possibly infinite) with $\operatorname{span}(S) = V$. Does there exist a subset of $S$ that is a basis for $V$?
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2answers
82 views

For vector x, y, when does |x+y| = |x|+|y|?

In general $|x+y|\le|x|+|y|$. When does equality hold? Spivak "Calculus on Manifolds" says the answer is not "when x and y are linearly dependent." However, that is the answer I get.
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1answer
64 views

Appending linearly independent vectors

Suppose i have a set of linearnly independent vectors v1,v2,v3 outside span[b1,b2] and i wanna check that $v_1,v_2,v_3,b_1,b_2$ forms a linearnly independent set. Can i do this mini-checkups of ...
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1answer
71 views

Dimension Recovery of $S \subset P_n(F)$

How is the subset of $P_n(F)$ consisting of all polynomials $f$ such that $f(1) = 0$ a subspace of $P_n(F)$? What is the dimension of this subset? Added from answer posted by Trancot on 18 Apr ...
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2answers
517 views

What does it mean that a finite set in a vector space has this property?

My homework problem says to let $S$ be a finite set in a vector space $V$ with the property that every $\vec x$ in $V$ has a unique representation as a linear combination of elements of $S$. Show that ...
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4answers
125 views

Dimensionality and Subspace Existence: A Potential Outlet for Disquisition

The subset of $F^n$ consisting of all vectors $(a_1,a_2,\dots,a_n)$ such that $a_1+a_2+\cdots+a_n=0$ is a subspace of $F^n$ and its dimension is ...(?).... Initially, my intuition said the ...
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1answer
46 views

On the Dimensionality of Space: An Elementary Analysis

The below theorem I am to prove. Perhaps you have a critique... Theorem 2.4 Let $W_1$ and $W_2$ be two subspaces of a vector space $V$. Then $\dim(W_1 \cap W_2)=\dim(W_1)$ if and only if $W_1 ...
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1answer
164 views

Kenneth Hoffman | Ray Kunze: An Inquiry into Symbolic Meaning

When those authors state the following $\bf{Theorem 6.}$ If $W_1$ and $W_2$ are finite-dimensional subspaces of a vector space $V$, then $W_1+W_2$ is finite-dimensional and \begin{eqnarray} \dim ...
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2answers
727 views

Is vector subtraction commutative?

Is Vector Subtraction commutative (a-b = b-a)? And if so how is it visually represented? My textbook states that it is, but I can't seem to figure out how to visually represent it with the ...
2
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1answer
179 views

Proving/disproving this is a linear subspace

I need to prove/disprove that $W$ is a linear subspace, and I'm not sure my approach is correct (especially the last point I'm making). Please correct me if I'm wrong. Let $V$ be a set of vectors ...
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4answers
240 views

Given two column vectors $a$ and $b$, what is the determinant of $A$ if $A=ab^T$

Given two column vectors $a$ and $b$ in $\mathbb R^n$ , $n \ge 2$, form the $n×n$ matrix $A = ab^T$. What is the determinant of $A$? (Hint: Examine linear dependence).
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1answer
97 views

Are coordinate projections continuous?

Okay I have been working under the assumption that this is "obvious" for a while now, but it started to bug me and now I'm fumbling to prove it. Suppose $X$ is a normed linear space (possibly ...
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1answer
136 views

On norm selection for the solution of an overdetermined linear system

I am considering the following linear system: $Ax = b$ Where: $A$ is $9000 \times 139$ $x$ is $139 \times 1$ and sparse $b$ is $9000 \times 1$ Most of the resources I have found online point to ...
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1answer
105 views

Summation and Subtraction of Vector Subspaces

Let $U, W$ be vector subspaces of V, such that $U, W \leq V$. Let $U-W = \left \{ u-w \;|\; u \in U, w \in W \right \}$ In this case, is $U+W = U-W$? I'm guessing yes because subtraction in this ...
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1answer
73 views

Pointed Convex cone: one-to-one correspondence extreme rays - extreme points

Hoi, let $V$ be a finite dimensional real vector space with inner product $\left\langle .\right\rangle$. Let $\Gamma\subset V$ and $\Gamma \neq \left\{0\right\}$ a pointed convex cone. (Pointed means ...
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2answers
90 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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1answer
32 views

Vector cross product

I have this question on my take home assignment and it is giving me a headache. Find the volume of the parallelepiped with three edges formed by <2,1,0> , <-1,2,0> and <1,1,2> using the ...
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1answer
279 views

Two linear functionals having the same Kernel are proportional

Let $V$ be a $k$-vector space, of finite dimension. Let $F,G:V\longrightarrow K$ be two non-zero $k$-linear applications. Suppose that $F$ and $G$ have the same kernel. Then $F$ and $G$ are ...
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1answer
309 views

Orthogonal complement of S in a finite field $\mathbb F_q$

For the following set S and corresponding finite field $\mathbb F_q$, fing the $\mathbb F_q$-linear span $\left<S\right>$ and its orthogonal complement $\left<S\right>$-perp. $$S = ...
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4answers
269 views

If $X$ is an orthogonal matrix, why does $X^TX = I$?

It's not immediately clear to me why this is true. My notes say that putting $n$ orthonormal vectors $ v_1, ..., v_n$ in the columns of $X$ gives $X^TX = I$, and it follows from this that the rows of ...
1
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1answer
878 views

Real and complex canonical forms of quadratic form

How do I find the canonical form of $$q_1(x,y,z)= 4x^2 +4xz+2yz$$ Now I have put it in matrix form as: $$\left( \begin{matrix} 4 & 0 & 2 \\ 0 & 0 & 1 \\ ...
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2answers
648 views

Basis of complex matrix vector space over $\Bbb{R}$

I understand that the basis of the vector space $$Mat_2(\Bbb{R}) = \begin{pmatrix}a_{11} & a_{12} \\ a_{21} & a_{22}\end{pmatrix}$$ over $\Bbb{R}$ is $$e = \left\{ \begin{pmatrix}1 & 0\\ 0 ...
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1answer
82 views

Inner product spaces that are isometrically isomorphic

I know this is a fundamental result in linear algebra, and although it is referenced in my textbook, it does not have a proof for it. I was wondering if someone could help me out: Let $V$ and $W$ be ...
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3answers
257 views

If $Ax=B$ has two solution, then there must be a third one?

How do I prove this conjecture? Let $A$ be a matrix, and $B$ be a column vectore. If $Ax=B$ has two solutions, then there must be a third one. Thanks in a advance!
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1answer
70 views

finding a suitable orthogonal matrix

Assume that $x,y \in \text{R}^n$ are two arbitrary vectors. Assuming that $x^\top y >0$, I want to prove that there exist an orthogonal matrix $U \in \text{O}(n)$ such that all elements of the ...
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1answer
189 views

Prove that D (the differential operator) maps V (a vector space) into V.

I'm quite confused about what "into" means here and, more importantly, how I am supposed to prove that something maps a vector space into (not onto) another vector space. Here's some of the ...
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1answer
79 views

What is the normal form for this line?

I have calculated the parametric form of a line as: $L = P_1 + tP_1P_3 = <2,2,0> + t<1,2,2>$. If I am given a point $ K = <1,-1,-1>$, how would I show the normal form of plane $E$ ...
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1answer
250 views

Proving linear independence

Let $A$ be an $n \times n$ matrix and suppose $v_1, v_2, v_3 \in \mathbb{R}^n$ are nonzero vectors that satisfy: $$ Av_1 = v_1 \\ Av_2 = 2v_2 \\ Av_3 = 3v_3 $$ Prove that $\{v_1, v_2, v_3\}$ is ...
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2answers
143 views

Linear algebra prove or disprove Kernel and Range

For a linear map $h: \mathbb R^3 \rightarrow \mathbb R^2$, the kernel of $h$ is a subspace of $ \mathbb R^2$. For a linear map $h: \mathbb R^3 \rightarrow \mathbb R^2$, the range of $h$ is a subspace ...
3
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5answers
236 views

Dimensions: $\bigcap^{k}_{i=1}V_i \neq \{0\}$

Let $V$ be a vector space of dimension $n$ and let $V_1,V_2,\ldots,V_k \subset V$ be subspaces of $V$. Assume that \begin{eqnarray} \sum^{k}_{i=1} \dim(V_i) > n(k-1). \end{eqnarray} To show that ...
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3answers
69 views

Linear algebra - how to tell where vectors lie?

I'm working my way (self-study) through Strang's text on Linear Algebra and am currently on Problem 1.2 #6. 6b) The vectors that are perpendicular to $V = (1,1,1)$ lie on a _ . 6c) The vectors that ...
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1answer
54 views

Kernels of Adjoints

Let $A$ be an $m \times n$ matrix. Show that $\mbox{Ker} A = \mbox{Ker} (A^*A)$. To do that you need to prove 2 inclusions, $\mbox{Ker} (A^*A)$ is a subset of $\mbox{Ker} A$ and $\mbox{Ker} A$ is a ...
3
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2answers
61 views

Generalised eigenvalue is eigenvalue if it is in the field

I would like to prove the following assertion: Let $\mathscr{F}$ be a field and $\mathscr{\phi}$ be an $\mathscr{F}$-linear endomorphism of a finite dimensional $\mathscr{F}$-vector space ...
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1answer
55 views

Determine Span of vectors?

I did not see question like this before? What is span of $(1,1+x,1+x+x^2,....,1+x+x^2+...+x^n)$ ? The question also says to let $V=P_n(X)$ be the space of all polynomials whose degrees are less than ...
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2answers
109 views

What is required to establish the law of cosines?

In my quantum computation course, we have been given nothing more than the basic axioms of a linear vector space, and and the properties of an inner product; but we have started referring to "the ...
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3answers
78 views

Counter Example: Span Inclusion

For any two subsets $S$ and $S'$ of a vector space $V$ does $span(S) \cap span(S') = span(S \cap S')$? If $S=ax, a \in \mathbb{R}$ in $\mathbb{R}^2$ and $S'=by, b \in \mathbb{R}$ in $\mathbb{R}^2$ ...
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6answers
205 views

Show that every subspace of $\mathbb{R}^n$ is a kernel of a linear map.

Let $S$ be a subspace of $\mathbb R^n$. Show that there is an $n \times n$ matrix $A$ such that $$S= \{x \in \mathbb R^n : Ax=0\}.$$ How to proceed?
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2answers
644 views

If a subset $S$ of a vector space $V$ is a subspace of $V$, then is $\langle S \rangle = S$?

I'm reading here on page 22 of Axler, Linear Algebra Done Right, where the following is stated: A $\bf{linear}$ $\bf{combination}$ of a list $(v_1,\dots,v_m)$ of vectors in $V$ is a vector of the ...
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1answer
55 views

How to show that $\mathrm{Sym}_{n\times n}(\Bbb{R})$ and $\mathrm{Skew}_{n\times n}(\Bbb{R})$ are subspaces of $\mathrm{M}_{n\times n}(\Bbb{R})$

A matrix $M \in \mathrm{M}_{n\times n}(\mathbb{R})$ is called symmetric (respectively, skew-symmetric) if $M^t = M$ (respectively, $M^t = -M$). How does one prove that the sets $\mathrm{Sym}_{n\times ...
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1answer
119 views

Vector Spaces Linear algebra

bI've been working through some problems in my Linear Algebra course and I've come across some that have me confused. I'm not particularly good at vector spaces so some help would be greatly ...
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2answers
388 views

Intersection | Subspaces | Span

If $W_1$ and $W_2$ are two subspaces of a vector space $V$, then $W_1+W_2$ is the intersection of all subspaces of $V$ that contain $W_1$ and $W_2$, right? Is the intersection of all subspaces of $V$ ...
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2answers
46 views

Picture of Space Layout

If $W_1+W_2$ is the intersection of all subspaces of $V$ that contain $W_1$ and $W_2$, then how should this be represented pictorially? Also, how do I prove that $W_1+W_2$ is the intersection of all ...
1
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2answers
1k views

Find an orthogonal vector to 2 vector

I have the following problem: A B C D are the 4 consecutive summit of a parallelogram, and have the following coordinates A(1,-1,1);B(3,0,2);C(2,3,4);D(0,2,3) I must find a vector that is ...