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
73 views

Coordinates relative to arbitrary 3D plane

Say that I have an arbitrary plane, $\mathcal{P}$, in $\mathbb{R}^3$ that is defined by a given vector, $\vec{v}_0$, on the plane and a normal vector, $\vec{n}$. I will be using $\vec{v}_{0}$ as a ...
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2answers
90 views

Angle between vectors of the form $(\cos A,\cos B,\cos C)$

The question: Two vectors $S=(\cos A,\cos B,\cos C)$, $S'=(\cos A',\cos B',\cos C')$, What is the angle between them? The answer is $\cos(\theta)$ = $\cos A.\cos A'+ \cos B.\cos B'+ \cos C.\cos C'$. ...
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2answers
60 views

Tensor Product problem.

Let $V$ be a vector space over $\Bbb F$, and let $x\not=0,y\not=0 $ be two elements in $V$. I want to show that $x\otimes_{_F} y=y\otimes_{_F} x$ iff $x=ay$ where $a\in \Bbb F$. I know the second ...
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1answer
45 views

Let p,q, r be fixed elements of a Field F

Let $p,q,r$ be the fixed elements of a field $F$ . Let $$W=\{(x,y,z) \in V^3(F)|px+qy+rz=0\}$$ Prove that $W(F)$ is a vector sub-space of $V^3(F)$ . I think i need to satisfy these conditions but ...
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3answers
288 views

Basis of Vector space $\Bbb C$ over rational numbers.

What will be the basis of vector space $\Bbb C$ over field of rational numbers? I think it will be an infinite basis! I think it will be $B=\{r_1+r_2i \mid r_1, r_2 \in \Bbb Q^{c}\}\cup\{1,i\}$. ...
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1answer
299 views

How to firgure out if a set of vectors represent lines, planes or hyperplanes?

if i am given a span of, let's say 3 vectors, what would be a way to determine if they represented a line, plane or a hyperplane? i have reduced siad vectors to reduced row echelon form, but don't see ...
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2answers
70 views

Is there an $x$ that can solve for this vector?

I'm trying to find an $x$ that solves for this:$$\left[\begin{matrix}1\\2\end{matrix}\right]+\left[\begin{matrix}x\\x\end{matrix}\right]=\left[\begin{matrix}2\\1\end{matrix}\right]$$ But I'm not sure ...
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0answers
99 views

Basis for an infinite dimensional vector space.

Is there any good paper that focus on the topic of basis for infinite dimensional vector space that I can read/ study. I found some papers that mention about this topic online, but they are very brief ...
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2answers
152 views

Prove the two subspaces are equals to ${F^3}$

$$\eqalign{ & W = \{ (0,b,c):b,c \in F\} \cr & U = \{ (a,a,a):a \in F\} \cr} $$ Prove that: $${F^3} = W \oplus U$$ A direction would be nice. Thanks
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1answer
100 views

Prove this is a bounded linear operator and find its operator norm?

I have a map $$A:(C[0,1], || \centerdot ||_\infty) \rightarrow \mathbb R, Ax = x(0) \forall x \in C[0,1]$$ and need to prove it's a bounded linear operator, and find its operator norm. I've tried ...
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1answer
43 views

Does there necessary exist a base of a vector space?

Given a vector space $V$, does there necessary exist a set $S$ of vectors from $V$ such that $S$ is a base of $V$?
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1answer
216 views

Show C(X) is a vector space over $\mathbb R$ with the following operations?

I have a set of continuous functions, $C(X): X \rightarrow R$ on a compact metric space, and definitions of addition & multiplication: $$(f+g)(x) = f(x)+g(x)$$ $$(\lambda f)(x) = \lambda ...
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4answers
454 views

Find a vector that is perpendicular to $u = (9,2)$

Attempt: We know perpendicular vectors have dot product $u \cdot v = 0$ therefore $[9,2] \cdot [x,y]$ = 0 $9x + 2y = 0$ what would I do now? thanks!
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0answers
95 views

Prove that a linear operator is indecomposable

Let $V$ be a fi nite-dimensional vector space over $F$, and let $T: V \rightarrow V$ be a linear operator. Prove that $T$ is indecomposable if and only if there is a unique maximal T-invariant proper ...
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1answer
79 views

Span of a f.g. $R$-module over the quotient field of $R$.

Let $R$ be an integral domain with quotient field $K$, $R \neq K$. Let $V$ be a finite dimensional vector space over $K$ and $M$ a finitely generated $R$-submodule of $V$. My question is how $KM=V$ is ...
0
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1answer
42 views

Orthogonal basis?

Theorem: Suppose $S$ is a subspace of $\mathbb{R}^n$, and vector $x \in \mathbb{R}^n$. If $x=s+t$, for $s \in S$ and $t \in S^{⊥}$, then $s$ and $t$ are unique. Proof: Suppose $x = s + t$ and ...
3
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1answer
67 views

Vector space $V$ over $\mathbb{Z}_p$ when $p$ is a prime

I need to determine if the following statement is true or false, if it's true, I need to prove it, else I need to give a counterexample: Let $V$ be a vector space over $\mathbb{Z}_p$ when $p$ is a ...
0
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1answer
52 views

Dot product and orthogonality?

Please only use the following definition of the dot product: $u \dot{} v = u^{T}v$ Using the above definiton only (not the cosine definition) why would the dot product being zero imply the angle ...
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3answers
1k views

Eigenvectors are linearly independent?

Theorem: Eigenvectors corresponding to distinct eigenvalues are linearly independent. Could someone give me a geometric interpretation of the theorem? Thanks!
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1answer
2k views

Vector line parallel to x-axis?

I answered the first part, but I don't the second part. Should the j and k vectors be zero?
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2answers
488 views

Geometric interpretation of the cofactor expansion theorem

I find the geometric interpretation of determinants to be really intuitive - they are the "area" created by the column vectors of the matrix. Could someone give me a geometric interpretation of the ...
0
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1answer
122 views

Affine Non Autonomous State Space system

Normally We all know the state space model of the the form der(x) = F*x(t)+G*u(t) y = H*x(t)+J*u(t). However I came across a state space model which has the following form der(x) = F*x(t)+G*u(t) + ...
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0answers
74 views

Find T cyclic operator that has exactly N distinct T-invariant subspaces

Let $T$ be a cyclic operator on $R^3$, and let $N$ be the number of distinct T-invariant subspaces. Prove that either $N$ = 4 or $N$ = 6 or $N$ = 8. For each possible value of $N$, give (with proof) ...
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2answers
117 views

Which of the following subsets of $\mathbb{R}^3$ are subspaces?

Which of the following subsets of $\mathbb{R}^3$ are subspaces? Explain the answer. $(a)\ \{(x,y,z) \mid x = 1\}$ $(b)\ \{(x,y,z)\mid z=3x−2y\}$ $(c)\ \{(x,y,z) \mid xy = 0\}$ $(d)\ \{(x,y,z)\mid ...
2
votes
1answer
58 views

Vectors and orthonormal matrix

For 2(a)(i), are the length of a =Sqrt(14) and b = sqrt(38)? For (ii), is the angle = 4.31? For (iii), is the answer 3.73? For (iv), is the answer -i+j+k? For (b) to (e), I have no idea what I ...
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3answers
60 views

Inverse of complex vector

How is the inverse of a complex vector calculated? In $\Bbb R$, the inverse vector $ X= \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \\ ...
1
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1answer
61 views

Vector equation in $\mathbb{R^3}$

Let $\vec{a}$ and $\vec{b}$ are non-zero vector in $\mathbb{R^3}$ which form an angle $30°$. Their length are $|\vec{b}|=\sqrt{3}|\vec{a}|$. How to solve the vector equation? $$ (\vec{x} \cdot ...
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2answers
44 views

Compute the linear transformation of a vector with respect to an ordered basis

I have a homework problem whose process I can't seem to figure out; any help provided is much appreciated: Let the representation of $L: R^3 -> R^2$ with respect to the ordered bases $S = {v_1, ...
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2answers
132 views

How to find dual spaces?

I would greatly appreciate it if you could kindly share how to find dual spaces? For example, let X be the vector space of n-dimensional vectors with the Euclidean norm. Prove that X*=X. I know a ...
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1answer
34 views

Vector space question with linear forms

If I have vector space of $n$ degree polynomials with real coefficients and $f(0) = 0$ and $f(1) = 0$, and inner product $\langle f,g \rangle = \int_0^1 f'(x) g'(x) dx$, how do I show that there is ...
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0answers
116 views

Inner product and canonical forms

If $V$ is a vector space with finite dimension and for some symmetric bilinear form $f: V \times V \rightarrow \mathbb{R}$, how to I show that $f$ defines an inner product iff the unique real ...
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1answer
79 views

Intuition behind independence of eigenvectors?

Theorem 6.14: Eigenvectors corresponding to distinct eigenvalues of A are linearly independent. My prof already gave us a proof of the theorem, so I'm not looking for another one. Could someone ...
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1answer
139 views

Find a basis and dimension of a vector space

Find a basis and the dimension of the vector space (on $R$) generated by $\lbrace u + v + w ; v + w + z ; w + z + u ; z + u + v \rbrace$, where $u,v,w,z$ are linearly independent vectors (from a ...
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1answer
28 views

$V=F_1^2$ over $F_2$: which operation to choose?

I don't understand the following vector space: $$V=\{(x,y)|x,y\in F_1\}$$ and $V$ is over field $F_2$, ($F_1$ is a field too). My question is: Is $V$ really a vector space? I am not talking about the ...
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1answer
223 views

Find transformation matrix with respect to a basis of an invariant subspace

Simple question but I've never encountered ones like that. $T$ is a linear transformation from $\mathbb R^3$ to $\mathbb R^3$ defined by: $T(v)=Av$ when $A=\begin{pmatrix} 1 & 2 & 2\\6 & ...
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2answers
85 views

Why do diagnolizable matrices have to be invertible?

My professor gave us this definition of a diagnolizable matrix. A matrix $A$ is diagnolizable if it's invertible and $$(Ax)_{\mathcal{B}} = Dx_{\mathcal{B}}$$ for some diagonal matrix $D$, basis ...
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1answer
53 views

Diagnolization of a non-invertible matrix?

Let $A$ be a $n \times n$ matrix. Let $\mathcal{B}$ be a basis for the subspace formed by the columns of $A$. Can there exist a diagonal matrix $C$ such that: $$Cx_{\mathcal{B}} = ...
3
votes
3answers
94 views

Do these matrices exist?

Say you have some non-zero vector in $x$. Can you have two matrices $A$ and $B$ such that: $$Ax = Bx$$ if $A$ and $B$ aren't the identity matrix? This isn't homework, just curiosity.
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4answers
145 views

Nullspace that spans $\mathbb{R}^n$?

My professor said that if for a $n \times n$ matrix $A$, $\text{null}(A) = \mathbb{R}^n$, then $A = 0_{n}$. Why is this true? I understand what its saying - if everything times this matrix is zero, ...
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1answer
82 views

Tensor product equivalent definitions

I'm studying tensor products right now and I've came across multiple definitions. The one I'm confused with is when we have vector spaces $V$ and $W$ and we define the tensor product as the quotient ...
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1answer
1k views

Show the W1 is a subspace of R4

I must prove that W1 is a subspace of $R^4$. I am hoping that someone can confirm what I have done so far or lead me in the right direction. $ W_1 = {(a_1,a_2,a_3,a_4) \in R^4 | 2a_1 - a_2 - 3a_3 = 0 ...
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2answers
887 views

Prove Convex Hull of Minkowski sum

I want to prove that the following holds, where the $+$ means Minkowski sum: $$ conv(A+B)=conv(A)+conv(B) $$ Let the convex hull of $A+B$ be $$ conv(A+B)=\sum_{j,k}\lambda_j\mu_k(a_j+b_k) $$ I ...
2
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0answers
158 views

Find all the invariant subspaces of T

T is a linear transformation, defined as the following: $T(p(x)) = xp(x)$, $T\colon R[X]\to R[X]$ Find all the invariant subspaces of $T$. As I see it, only the trivial subspaces $0$, $R[X]$ are ...
2
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1answer
34 views

Relationships of orthogonal subspaces

I'm struggling with the following problem: Let $U_1$ and $U_2$ be supspaces of $V$. What is the relation between $(U_1+U_2)^\perp$ and $U_1^\perp \cap U_2^\perp$; and between $(U_1 \cap U_2)^\perp$ ...
1
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1answer
45 views

Vector subspace clarification

We need to check if the following sets are vector subspaces: $$ S_1=\{(0,0),(1,2),(2,1) \};V=(\Bbb Z_3)^2 \\ S_2=\{P\mid\exists x \in \Bbb R :P_{(x)}=0 \};V=\Bbb R_n [x]$$ For $S_1$ I just add: ...
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0answers
56 views

What is first edge position in the Minkowski sum of two convex polygons in the plane?

I am trying to understand the informal algorithm of the Minkowski sum of two convex polygons in the plane as described here: ...
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1answer
20 views

find examples to prove $ A \cup B $ is not part of the $ V $

For $ A, \: B $ is the subspace of $ V $ Find examples to prove $ A \cup B $ not a subspace of $ V $ I learn about this program should not know how. Desire to help people and give a solution ...
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1answer
55 views

Conditions for linear independence of extended vector systems

Assume $$g: R^n \times R^m \rightarrow R^n$$ $$h: R^n \times R^m \rightarrow R$$ $$(x,y) \in R^n \times R^m$$ I would like to show that the following vectors are linearly independent: ...
0
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2answers
714 views

Finding vector and parametric equations provided only one point.

Normally to answer these questions I have a point and one or two vectors. However, for this one I only have a point. How can I concoct these equations provided there is limited information? Find ...
0
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1answer
56 views

finding projection on subspace

I have a question: Find the projection of $v = <1,2,1>$ on $span(<3,1,2>,<1,0,1>)$ in $R^3$ calling the vectors in the span a and b $$proj_w V = \frac{V \cdot a }{a^2} ...