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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Proving replacement theorem?

I want to see if I am understanding the proof of the replacement theorem correctly. Let $V$ be a vector space that is spanned by a set $G$ containing $n$ vectors. Let $L \subseteq V$ be a linearly ...
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5answers
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Is closure of convex subset of $X$ is again a convex subset of $X$?

Today I was going through my text book exercise and got hold of the following question. Let $X$ be a normed linear space with norm $\lVert\cdot\rVert$ and $A$ is a non empty convex subset of $X$ ...
4
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3answers
324 views

Embedding torsion-free abelian groups into $\mathbb Q^n$?

Glass' Partially Ordered Groups states without proof: Every torsion-free abelian group can be embedded into a rational vector space (as a group). Can someone link me to a proof of this? It ...
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1answer
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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 = \begin{...
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1answer
47 views

If a linear transformation $T$ has $z^n$ as the minimal polynomial, there is a vector $v$ such that $v, Tv,\dots, T^{n-1}v$ are linearly independent

Let $T: V \to V$ with the minimal polynomial $z^n$. Prove that there's a vector $v$ such that $v, Tv, T^2v, ..., T^{n-1}v$ are linearly independent. The way I did it orginally was not allowed. No ...
3
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3answers
219 views

Prove that $\mathbb{Z}$ is not isomorphic to additive group of any vector space over any field.

Prove that $\mathbb{Z}$ is not isomorphic to additive group of any vector space over any field. Proof. Surpose that: $\phi : (A, +) \rightarrow \mathbb{Z} $ is an isomorphism. Then there is some $r∈...
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5answers
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A linear transformation $T:V\to V$ is one-to-one if and only if it is onto

Let $V$ be a finite dimensional vector space. Show that a linear transformation $T\colon V \to V$ is one-to-one if and only if it is onto. The hint I was given was that one only needs to show that $...
2
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1answer
924 views

Could intersection of a subspace with its complement be non empty.

If that is possible could you please correct my understanding about complement of a subspace. From what i recall from set theory. A complement of a set B is the set U - B where U is the universal ...
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3answers
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The kernel and image of $T^n$

I need help with this question: Let $V$ be a finite vector space where $ \dim V = n $, over the complex numbers and let $ T: V\to V $ be a linear transformation. Prove that $ V = \ker(T^n) \oplus Im(...
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4answers
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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 $$ a_{ij}=\pmatrix{\...
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2answers
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Let $V$ be a $K$-vector space, $f: V \rightarrow V$ a linear map. Under some condition, show that $v, f(v),…,f^n(v)$ are linearly independent. [duplicate]

Let $V$ be a $K$-vector space, $f: V \rightarrow V$ a linear map. Let $v \in V$. May a number $n ≥ 0$ exist, so that: $f^n(v) \not= 0$ and $ f^{n+1}(v) = 0$. Show that $v, f(v),...,f^n(v)$ are ...
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4answers
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Basis of a $2 \times 2$ matrix with trace $0$

I have a question that I do not understand and it goes like this: Find a basis for the set $W$ of all matrices A in $M_{2\times2}$ with trace $0$: i.e. all matrices $$ \begin{pmatrix} a & b\\ c ...
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2answers
790 views

Non-degenerate symmetric bilinear form; dimension formulae.

Let $E$ be a vector space endowed with a non-degenerate symmetric bilinear form. Show $\dim F+\dim F^{\perp}=\dim E=\dim\left(F+F^{\perp}\right)+\dim\left(F\cap F^{\perp}\right)$ Lang uses this ...
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2answers
399 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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3answers
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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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votes
3answers
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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 $pq=...
7
votes
3answers
254 views

Examples of 'almost' vector spaces where unitary law fails

I was looking at the definition on wikipedia of a vector space (similar/equivalent definitions are everywhere, but I thought I'd list it here for completion): A vector space over a field $F$ is a set ...
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3answers
132 views

The basis of a matrix representation

If I have the linear map $f:\Bbb{R}^n\rightarrow \Bbb{R}^m$ then we can write $f$ as like the following: $$f\left(\vec x\right)=A\vec x$$ Where $A$ is a matrix. I think $A$ is called the standard ...
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5answers
188 views

Minkowski sum of two disks

An open disk with radius $r$ centered at $\mathbf{p}$ is $D(\mathbf{p}, r)=\{\mathbf{q} \mid d(\mathbf p, \mathbf q) < r\}$, and the Minkowski sum of two sets $A$ and $B$ is $A \oplus B=\{\mathbf p ...
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3answers
394 views

A vector space over an infinite field is not a finite union of proper subspaces?

Show that if $V$ is a vector space over an infinite field $\mathbb{F}$, then $V$ cannot be written as set-theoretic union of a finite number of proper subspaces.
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Is null vector always linearly dependent?

I'm trying to find the column space of $\begin{bmatrix}a&0\\b&0\\c&0\end{bmatrix}$, which I think is $span\left(~\begin{bmatrix}a\\b\\c\end{bmatrix}~\begin{bmatrix}0\\0\\0\end{bmatrix}~\...
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5answers
191 views

What is $\Bbb{R}^n$?

I earlier asked this question The basis of a matrix representation. I now have a another question related to the same topic. The vector space $\Bbb{R}^n$ I have seen defined as all $n$-tuples of real ...
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870 views

A theorem concerning unique linear mapping between vector spaces: What does it say?

Theorem (from Schaum's Linear Algebra) Let $V$ and $U$ be vector spaces and $\{v_1, \ldots, v_n\}$ be a basis on $V$. Let $\{u_1,\ldots, u_n\}$ be arbitrary vectors in $U$. Then there exists a unique ...
3
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1answer
149 views

How many (unordered) bases does $\Bbb F_q^n$ have as a vector space over $\Bbb F_q$?

Following the recommendation here to get this question out of the unanswered queue, I've changed this from a proof-verification question into an answer-your-own. Here's the question again in case ...
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5answers
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Proof of uniqueness of identity element of addition of vector space

Please rate and comment. I want to improve; constructive criticism is highly appreciated. Please take style into account as well. Proof of uniqueness of identity element of addition of vector space ...
3
votes
1answer
4k views

Subspaces of all real-valued continuous functions on $\mathbb{R}^1$

I'll go ahead and give you the problem first, and then explain my trouble with it. Which of the following subsets are subspaces of the vector space C(-$\infty$,$\infty$) defined as follows: Let V be ...
2
votes
2answers
246 views

Find a matrix such that $Ax=0$

Let $$W = span\left\{ {\left( {\matrix{ 1 \cr 0 \cr 0 \cr 1 \cr } } \right),\left( {\matrix{ 0 \cr 2 \cr 1 \cr { - 1} \cr } } \right)} \right\}$$ I was asked ...
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2answers
364 views

Every subspace of a vector space has a complement

I want to see if my proof is true or I thought very trivially? If $H$ is a subspace of a finite dimensional vector space $V$, show there is a subspace $K$ such that $H\cap K=0$ and $H+K=V$ So ...
2
votes
3answers
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Proof if $a \vec v = 0$ then $a = 0$ or $\vec v = 0$

I'm kicking myself over this one, but I just can't seem to make the argument rigorous. From Axler's Linear Algebra Done Right: for a vector space $V$ with an underlying field $F$: Take an element $a$...
2
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1answer
325 views

Are all fields vector spaces?

Are $\mathbb{Z_p},\mathbb{Q},\mathbb{R},\mathbb{C}$ above themselves vector space? Is a field above anoother field a vector space? As for 1. we know that $\Bbb R^n$ is a vector space so in ...
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1answer
264 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 ...
2
votes
2answers
208 views

Suppose that $V$ is a vector space, and $W$ is a subspace of $V$. If $V$ is finite dimensional, then prove $W$ too must be finite dimensional.

Suppose that $V$ is a vector space, and $W$ is a subspace of $V$. If $V$ is finite dimensional, then prove $W$ too must be finite dimensional. It seems intuitively obvious that the dimension of $W$...
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2answers
49 views

Do the spaces spanned by the columns of the given matrices coincide?

Reviewing linear algebra here. Let $$A = \begin{pmatrix} 1 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 1 \end{pmatrix} \qquad ...
2
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1answer
64 views

Using SVDs to prove $C(XX^{\prime}) = C(X)$

Let $C$ denote the column space. I would like to prove $C(XX^{\prime}) = C(X)$ for $X \in M_{n \times p}$, $X^{\prime}$ denoting the transpose of $X$. This answer suggests using singular value ...
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1answer
128 views

Distance of point for a set in linear spaces

Let $X$ a normed linear spaces, $Y \subset X$ a subspace and $z \in X$ an arbitrary point. How can we show that: $$\text{dist} (z, Y) = \sup \{\psi(z) \ | \ \|\psi\| = 1, \psi \equiv 0 \ \text{on} \ ...
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1answer
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How to find the gradient for a given discrete 3D mesh?

I have a 3D mesh that is looking like this: ie I have a set of triangles in a 3D space, and they are all linked by their edge. I have to compute the gradient associated with this field, at each edge ...
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3answers
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Finding the Angle theta between two 2d vectors.

Find the angle α between the vectors $$\begin{bmatrix}3 \\ 5 \end{bmatrix}and \begin{bmatrix}-2 \\ 3 \end{bmatrix} $$ I found 64.654, but apparently is wrong from what the webwork says. Can anyone ...
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1answer
221 views

Counter example for a result of intersection of subspaces

I am struggling with this question from Halmos's text, please ignore the imperative language. "Suppose that $L, M$ and $N$ are subspaces of a vector space. Show that the equation $$L \cap (M + N) ...
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435 views

First Order Language for vector spaces over fields

I'm attempting to come up with a first order language $L$ that is able to describe vector spaces over fields. I came up with a few sets of nonlogical symbols. $Rs_L=\{Scal,Vec\}$ where $Scal,Vec$ are ...
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1answer
87 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 2013:...
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$T\circ T=0:V\rightarrow V \implies R(T) \subset N(T)$

Question Let $T:V \rightarrow V$ be a linear map. How do I prove that $T \circ T = T_0$ ( the zero linear map) iff $R(T) \subset N(T)$? Attempt \begin{eqnarray} T\circ T=T(T(v))&=&T(T(v-0))\...
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1answer
117 views

Is $\mathbb{R^Z}$ or its elements countable?

Continue on the self study on infinite vector spaces. According to this link, $\mathbb{R^Z}$ has elements of the following form: $$(y_k)_{k\mathbb{\in Z}}=(\dots y_{-1},y_0,y_{1}\dots)$$ Or more ...
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1answer
69 views

Linear basis of sum of kernels of two linear applications from $\mathbb R^4$ to $\mathbb R^2$

Let $$L_{1}(x_{1},x_{2},x_{3},x_{4})=(3x_{1}+x_{2}+2x_{3}-x_{4}, 2x_{1}+4x_{2}+5x_{3}-x_{4})$$ and $$L_{2}(x_{1},x_{2},x_{3},x_{4})=(5x_{1}+7x_{2}+11x_{3}+3_{4}, 2x_{1}+6x_{2}+9x_{3}+4x_{4})$$ Let $U_{...
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83 views

Unitary map between sets of vectors

Suppose I have two sets of vectors, $E_1=\{v_i\}_{i=1}^{k}$ and $E_2=\{u_i\}_{i=1}^{k}$, with each vector belonging to $\mathbb{C}^k$. When is it possible to find a unitary matrix that maps $E_1$ to $...
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3answers
78 views

Finding a basis for a certain vector space of periodic polynomials

I am having a little bit of trouble solving an homework question. I found that $S={ p(x) \in R_4[x]} \big| p(x)=p(x-1) $ is a vector space. Now I need to find some set, K that holds ${span(k)=S}$ ...
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2answers
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Calculate intersection of vector subspace by using gauss-algorithm

There are two vector subspaces in $R^4$. $U1 := [(3, 2, 2, 1), (3, 3, 2, 1), (2, 1, 2 ,1)]$, $U2 := [(1, 0, 4, 0), (2, 3, 2, 3), (1, 2, 0, 2)]$ My idea was to calculate the Intersection of those two ...
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2answers
149 views

Calculating new vector positions

I'm using the following formula to calculate the new vector positions for each point selected, I loop through each point selected and get the $(X_i,Y_i,Z_i)$ values, I also get the center values of ...
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0answers
326 views

Another proof of uniqueness of identity element of addition of vector space

Please rate and comment. I want to improve; constructive criticism is highly appreciated. Please take style into account as well. The following proof is solely based on vector space axioms. Axiom ...
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3answers
59 views

How to determine if two points lie on a vector, given a unit vector

If you have two points, $A$ and $B$, at $(1,1,1)$ and $(1,1,7)$ respectively, and a unit vector $C (0,0,1)$. What's a way to find if unit vector C, will cross B if C extends forever. (Unit vector C ...
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2answers
373 views

Lower bound involving the rank of the composition of linear transformations

The following question is about a lower bound on the rank of a composition of functions given as a simple expression for the two terms of the sum involved in the inequality. Consider finite-...