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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55 views

Is this statement about vectors true?

If vectors $A$ and $B$ are parallel, then, $|A-B| = |A| - |B|$ Is the above statement true?
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119 views

incident angles between rays, falling on an oblique plane

I am having really two simple questions, but following two things are confusing me. Question 1 If I know plane parameter (v3) of a given plane (say AB); if a pair of rays are incident at a ...
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3answers
196 views

Invertibility in a finite-dimensional inner product space

Let $T$ be an invertible linear operator on a finite-dimensional inner product space. I just want a hint as to how I should prove that $T^{\star}$ is also invertible and $( T^{-1} )^{\star} = ( ...
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2answers
69 views

Am I understanding vectors and matrices properly?

So, here is my understanding of a Vector: A vector is an ordered set of real numbers that lie in the space $R^n$ where $n$ is the size of the vector. So if ...
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4answers
368 views

What is the subspace of $\mathbb{R}^3$ generated by $v_1= (2, -1,1)$ and $v_2= (1,2,3)$

What is the subspace of $\mathbb{R}^3$ generated by $v_1=(2, -1,1)$ and $v_2=(1,2,3)$? my options: $[\vec v_1,\vec v_2]=\{(x,y+1,x-y); x,y\in\mathbb R\}$ $[\vec v_1,\vec v_2]=\{(x,y,x+y); ...
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1answer
51 views

If $M$ is invariant under $L$ is it a vector space

I'm reading Linear Algebra by Peterson, and the excercise on p.31 reads as follows: Let $L: W \to V$ be a linear operator and $V$ a vector space over $\mathbb{F}$. Show that if $M \subset V$ is ...
4
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1answer
56 views

Given $T(A) = A^t$ in $M_{n\times n}(\mathbb R)$. Find the polynomials and find if it's diagonalizable

Given the vector space $M_{n\times n} (\mathbb R)$ and a transformation $T(A) = A^t$ (transpose): Find $m_T$, $P_T$ (the minimum polynomial and the characteristic polynomial respectively.) ...
2
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1answer
298 views

Show that if $S,T$ are linear transformations $N(ST) \leq N(S) + N(T)$

I have a finite dimensional vectors space $V$, and two linear transformations on $V$, $S$ and $T$. I need to show that $N(ST) \leq N(S) + N(T)$. Can anybody spot what is wrong with this argument: ...
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4answers
146 views

If $T\colon V\to V$ is linear then$\text{ Im}(T) = \ker(T)$ implies $T^2 = 0$

I'm trying to show that if $V$ is finite dimensional and $T\colon V\to V$ is linear then$\text{ Im}(T) = \ker(T)$ implies $T^2 = 0$. I've tried taking a $v$ in the kernel and then since it's in the ...
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1answer
62 views

Show that there is a subspace $W^\prime$ of $V$ such that $W \cap W^\prime = U$ and $W + W^\prime = V$.

Let $U$ and $W$ be subspaces of $V$ with $U \subseteq W$. Show that there is a subspace $W^\prime$ of $V$ such that $W \cap W^\prime = U$ and $W + W^\prime = V$. I'm not quite sure where to begin ...
2
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1answer
214 views

Prove that V contains a subspace of each dimension

Let $V$ be an $n$-dimensional vector space over $\mathbb R$. Prove that $V$ has a subspace of dimension $r$ for each $0 \le r \le n$. Is this as simple as saying that $V$ has a basis of $n$ elements, ...
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1answer
41 views

Understanding a formula on acceleration

We are given the formula in vectors, $\underline{a}=(μ\underline{r} \times \underline{v})/r^3$ where "$\underline{a}$" is the acceleration of any particle, "$\underline{r}$" is the displacement and ...
2
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3answers
333 views

$W_1^\perp + W_2^\perp = (W_1 \cap W_2 )^\perp$: Can a set be a function? Can two such “functions” be composed?

How do I prove the following proposition: $$W_1^\perp + W_2^\perp = (W_1 \cap W_2 )^\perp$$ Note that there have been other questions (here, here) just asking about the $\subseteq$ inclusion, so ...
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1answer
52 views

Can't figure out this transformation matrix

So basically I want to write a transformation matrix to take me out of one coordinate system and into another. The transformation has to be as follows: 1) The positive z axis normalized as ...
2
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4answers
374 views

Proof: $\det\pmatrix{\langle v_i , v_j \rangle}\neq0$ $\iff \{v_1,\dots,v_n\}~\text{l.i.}$

Let $V$ be a real inner product space and $S=\{v_1,v_2, \dots, v_n\}\subset V$. How am I to prove that $S$ is linearly independent if and only if the determinant of the matrix $$ ...
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0answers
64 views

Derivative of a vector with scalar product in denominator

I'm struggling with a partial derivative of the following form: \begin{equation} \frac{\partial}{\partial \vec{x}} \frac{\vec{x}}{\vec{e}_3^T\,\vec{x}}, \end{equation} where $\vec{x} \in ...
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1answer
117 views

Hermitian Inner Product | Basis | Orthogonal Complement

If I say $X = \{x, x'\}\subset\mathbb{F}^3$ is a subspace, where $x$ and $x'$ are linearly independent (for some field $\mathbb{R}$ or $\mathbb{C}$), with $$\mathbb{F}^n := ...
3
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1answer
80 views

Hamiltonian Quaternions: A Call for Counterexample for ($AB=I_m \implies n≥m$)

How can I build a counterexample to $AB=I_m \implies n≥m$ in the ring of Hamiltonian quaternions? Notice that the vectors $(1,i)$ and $(j,k)=(1,i)j$ are linearly dependent as vectors of the right ...
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1answer
486 views

how to solve vectors in 3D?

We had our examination a few days ago. I was not able to answer this question. So I decided to remember it and answer at home but still I can't answer this on my own. Please help me to answer this ...
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2answers
279 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 ...
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1answer
561 views

Cyclic vector space

In class we defined what it means that there is a creating element $v$ of a vector space, such that for an endomophism $A$ on $V$ we have: ${\rm span}(v,Av,...,A^{n-1}v)=V$. Also we said that if the ...
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2answers
352 views

Finding a basis with Change of Basis

Find the coordinate vector for v relative to the basis S = {v1, v2, v3} for $R^3$. $$v = (2,-1,3);$$ $$v1 = (1,0,0); v2 = (2,2,0); v3 = (3,3,3);$$ So I did and I got the coordinate vector ...
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2answers
502 views

Matrix representation of the adjoint of an operator, the same as the complex conjugate of the transpose of that operator?

Since I'm not taking summer classes I decided to do some self learning on more advanced mathematics, and I've found myself stuck on this problem: I have to show that for any operator $\hat{A}$ the ...
2
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3answers
466 views

How to prove $\lvert \lVert x \rVert - \lVert y \rVert \rvert \overset{\heartsuit}{\leq} \lVert x-y \rVert$?

I'm trying to show that $\lvert \lVert x \rVert - \lVert y \rVert \rvert \overset{\heartsuit}{\leq} \lVert x-y \rVert$. A hint would be nice.
3
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3answers
112 views

Multipliciousness within an inner product space.

Question: Let $V$ be an inner product space and $v,w\in V$. Prove that $\lvert\langle v,w\rangle\rvert=\lVert v\rVert \lVert w\rVert$ if and only if one of the vectors $v$ or $w$ is a multiple of ...
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3answers
308 views

Prove $p^2=p$ and $qp=0$

I am not really aware what's going on in this question. I appreciate your help. Let $U$ be a vector space over a field $F$ and $p, q: U \rightarrow U$ linear maps. Assume $p+q = \text{id}_U$ and ...
1
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2answers
247 views

Space spanned by matrices

I have a set of 5 by 5 matrices, M1,M2,...,M19 ,M20. I want to try to find a basis from this set and also to find relationships between these matrices. This is how I think I should approach the ...
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6answers
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A basis for a subspace of $\mathbb R^3$

I have the following question: Find the basis of the following subspace in $\mathbb R^3$: $$2x+4y-3z=0$$ This is what I was given. So what I have tried is to place it in to a matrix ...
6
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1answer
124 views

Tangential Space of a differentiable manifold is always $\mathbb R^n$?

Let $\mathcal M$ be a differential manifold with a point $p$. Let U be an open set, $p\in U$, on $\mathcal M$ and let $\phi,\psi:U\to \mathbb R^n$ be a charts on $\mathcal M$. I'm having diffculties ...
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1answer
202 views

REVISITED$^1$: How is the determinant of a matrix $A\in M_{2\times 2}(\mathbb{R})$ considered a bilinear form?

I'm trying to prove that $B(X,Y)=\det (X+Y) - \det (X) - \det (Y)$ is a blinear form on the vector space $A$ is from, and also trying to determine if it is an inner product space. I think if I know ...
1
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1answer
71 views

What's the dimension of this vectorspace?

Consider the concatenation operator, e.g. $(1,2)+(3,4)=(1,2,3,4)$. With the conceit that adjacent numbers can "cancel out", e.g. $(1,2)+(-2,3)=(1,3)$, this is a group. We can make it a vectorspace by ...
3
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0answers
212 views

How to show annihilator has dimension m-n (with Proof)

I would like to show the following: Given a vector spaces $V$, a subspace $S \subset V$ and an the dual space $V^*$ to $V$. Show that: $$\dim(N)+\dim(S) = \dim(V) = \dim(V^*)$$, where $N \subset V^*$ ...
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0answers
148 views

Curl in cylindrical coordinates

I'm trying to figure out how to calculate curl ($\nabla \times \vec{V}^{\,}$) when the velocity vector is represented in cylindrical coordinates. The way I thought I would do it is by calculating: ...
1
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2answers
44 views

If $b = c \times a$ and $c = a \times b$, and length $b$ = length $c$, $a$ is a unit vector.

If $\vec b = \vec c \times \hat a\,$ and $\,\vec c = \hat a \times \vec b\,$, and $|\vec b|$ = $|\vec c|$. Assuming $\vec b \ne 0$. I have managed to prove $\vec a$, $\vec b$ and $\vec c$ are ...
2
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1answer
149 views

This is regarding vector space in linear algebra…

Let $V$ be the set of all positive real numbers.On $V$ define the addition and the scalar multiplication in the following manner. addition: $x+y = xy$ scalar multiplication : $rx= x^r$ Prove or ...
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2answers
149 views

Why does $v \in \ker(A- \lambda I)^m$ imply that $Av$ is contained in $\ker(A- \lambda I)^m$?

Given an $n$ dimensional vector space $V$ and an $n \times n$ matrix $A$, Then why does $v \in \ker(A- \lambda I)^m$ imply that $Av$ is contained in $\ker(A- \lambda I)^m$? I'm not seeing it.
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3answers
252 views

What is the relationship between $(u\times v)\times w$ and $u\times(v\times w)$?

Given three vectors $u$, $v$, and $w$, $(u\times v)\times w\neq u\times(v\times w)$. This has been a stated fact in my recent class. But what is the ultimate relationship between them? I would presume ...
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3answers
2k views

which vectors are perpendicular to each other?

which vectors are perpendicular to each other? $a = (1, -2, 3)$, $b = (5, 4, 1)$, $c = (1, 0, -5)$ Do i just take the dot product of 2 of them. If the dot product they are at $90^\circ$? But how do ...
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2answers
216 views

Cyclic vectors in a real vector space

Let $V$ be an n-dimensional vector space over $\mathbb{R}$ and $T:V \rightarrow V$ be linear. Call a vector $v \in V$ cyclic if $V$ is spanned by $\{v, \ Tv, \ T^2v,..\}$. Question: Show that $v$ is ...
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1answer
265 views

Finding torque of a force [closed]

A force $\vec F=2\hat i+3\hat j$ is acting on a point $(1,0,3)$. Find the torque of this force about the axis along $y=3x$. I know 3-D geometry. So, I can handle equations of lines in 3-D too. ...
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1answer
51 views

Show that a certain map $V \to \mathbb R$ is an element of the dual Space $V^*$

Given a vector space $V = \operatorname{Mat}_{2\times2}(\mathbb{R})$. Define $\varphi:V\to\mathbb{R}$ by $$\varphi\left(\begin{pmatrix}a & b \\c & d\end{pmatrix}\right) = 2a+b$$ Prove that ...
0
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1answer
56 views

Inner product Proof,

So $V$ is an inner product space (finite dimensional) with inner product defined. If $v$ and $w$ are vectors in $V$, how would one go about proving this? $\langle \phi_\beta (x), \phi_\beta (y) ...
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2answers
232 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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3answers
2k views

Show that the area vectors for a general tetrahedron sum to zero

Using vector addition and multiplication, it is possible to show that the sum of the area vectors for a general closed tetrahedron in $\mathbb{R}^3$ (3-space) is zero. Hint: start by writing down ...
2
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2answers
109 views

Are there nonlinear operators that have the group property?

To be clear: What I am actually talking about is a nonlinear operator on a finitely generated vector space V with dimension $d(V)\;\in \mathbb{N}>1$. I can think of several nonlinear operators on ...
3
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3answers
158 views

How to prove that sequence spaces $l^{p}, l^{\infty}$ and function space $C[a. b]$ are of infinite dimension

I am studying about the sequence space $l^{p}, l^{\infty}$ and function space $C[a. b]$. It is mentioned in the book that all of these spaces are of infinite dimension. I want to prove that these ...
3
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1answer
531 views

Basis of vector fields on manifold

For a typical real manifold of $n$ dimensions, the basis of a tangent vector space $T$ at point $p$ is $$\frac{\partial}{\partial x_i},\ldots,\frac{\partial}{\partial x_j}$$ So is the general basis of ...
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2answers
50 views

Curve defined by a vector

http://i.stack.imgur.com/tD4Bn.png I'm studying line integrals with a curve as a vector, but I couldn't understand the 'dr' part. First of all: the curve isn't really a curve, it's like some points ...
2
votes
1answer
88 views

Convergence of coordinates to zero

Consider a normed finite-dimensional vector space $V$ with some norm $|| .||$ Say a sequence of vectors in this vector space $v_m \rightarrow 0$ where $0$ is the zero vector. Let ...
7
votes
3answers
351 views

Why define vector spaces over fields instead of a PID?

In my few years of studying abstract algebra I've always seen vector spaces over fields, rather than other weaker structures. What are the differences of having a vector space (or whatever the ...