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 ...

learn more… | top users | synonyms

10
votes
5answers
15k views

How to solve this to find the Null Space

What I did : I put this into reduced row echleon form: $$\begin{bmatrix} 1 & -2 & 2 & 4 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 ...
7
votes
2answers
6k views

Span of an empty set is the zero vector

I am reading Nering's book on Linear Algebra and in the section on vector spaces he makes the comment, "We also agree that the empty set spans the set consisting of the zero vector alone". Is Nering ...
5
votes
2answers
83 views

Necessary condition for have same rank

Let $P,Q$ real $n\times n$ matrices such that $P^2=P$ , $Q^2=Q$ and $I-P-Q$ is an invertible matrix. Prove that $P$ and $Q$ have the same rank. Some help with this please , happy year and thanks.
4
votes
2answers
57 views

What should we understand from the definition of orthogonality in inner product spaces other than $\mathbb R^n$?

In the beginning of linear algebra courses, there are vectors in $\mathbb R^n$ and the dot product is introduced. We learn that if the dot product of two vectors is zero, then these vectors are called ...
4
votes
3answers
314 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 ...
4
votes
1answer
2k views

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 ...
4
votes
3answers
117 views

For subspaces, if $N\subseteq M_1\cup\cdots\cup M_k$, then $N\subseteq M_i$ for some $i$?

I have a vector space $V$ over a field of characteristic $0$. If $M_1,\dots,M_k$ are proper subspaces of $V$, and $N$ is a subspace of $V$ such that $N\subseteq M_1\cup\cdots\cup M_k$, how can you ...
4
votes
1answer
2k views

Grassman formula for vector space dimensions

If $U$ and $W$ are subspaces of a finite dimensional vector space, $$ \dim U + \dim W = \dim(U\cap W) + \dim(U + W)$$ Proof: let $B_{U\cap W} = \{v_1,\ldots,v_m\}$ be a base of $U\cap W$. If we ...
3
votes
3answers
215 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 ...
3
votes
6answers
747 views

Nilpotent linear operators

Suppose that $T : V \to V$ is a linear operator on an $n$-dimensional vector space $V$. (a) Show that for all $i$, $\ker T^i \subset \ker T^{i+1}$. (b) Show that if $\ker T^k = \ker ...
3
votes
5answers
3k views

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 ...
3
votes
1answer
1k views

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 = ...
3
votes
3answers
514 views

Algebra: Orthogonal Complement

Problem Let $V$ be a real inner product space and $U \subset V$. Show that $(U^{\perp})^{\perp}=U$. Progress Clearly for $x\in U$ we have that $\langle x,v \rangle=0$ for all $v \in ...
3
votes
3answers
4k views

Geometric interpretation of the multiplication of complex numbers?

I've always been taught that one way to look at complex numbers is as a cartesian space, where the "real" part is the x component and the "imaginary" part is the y component. In this sense, these ...
2
votes
3answers
850 views

Sum of closed subspaces of normed linear space

Problem Suppose $R$ is a normed linear space, then show that: If $M$ is closed subspace of $R$ and $N$ a finite dimensional subspace of $R$, then the set $$M+N=\{ z : z = x + y , x \in M , y \in N ...
2
votes
4answers
759 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 $$ ...
1
vote
2answers
55 views

Let $V$ be a $K$ - Vector Space, $f: V \rightarrow V$ a linear map. Under some condition, show that $v, f(v),…,f^n(v)$ is linear independent.

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 $v, f(v),...,f^n(v)$ is ...
1
vote
2answers
119 views

About definition of “direct sum of $p$-vector subspaces”

In the books 1 and 2, in Somme directe d'une famille de sous-espaces vectoriels, I am reading the following: 1) let $E,F$ two vector subspaces of $V$, $E+F$ is direct sum, $E+F \doteq E\oplus F$, if ...
1
vote
3answers
992 views

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 ...
1
vote
1answer
566 views

What does this linear algebra notation mean?

I'm trying to prove that a particular $V$ is a $\Bbb{Q}$-vector space. The question says to take the element $0_V = 1$, the function $+_V : V \times V \to V$ given by the function $[x +_V y = xy]$, ...
1
vote
2answers
379 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 ...
1
vote
4answers
5k views

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 ...
0
votes
3answers
4k views

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.
7
votes
3answers
236 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 ...
7
votes
3answers
678 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 ...
5
votes
3answers
131 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 ...
4
votes
5answers
180 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 ...
4
votes
3answers
314 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} = ( ...
3
votes
2answers
105 views

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 ...
3
votes
5answers
629 views

If $n=\dim(V)$ and $n$ vectors are linearly independent, then they form a basis

If $V$ is a vector space and $v_1, v_2, . . . , v_n \in V$ span $V$, and $u_1, u_2, . . . , u_m ∈ V$ are linearly independent, then $m\le n$. Use this to prove that if $V$ has dimension $n$ and $u_1, ...
3
votes
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 ...
3
votes
5answers
2k views

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
148 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 ...
3
votes
2answers
790 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 ...
3
votes
1answer
3k 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
168 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 ...
2
votes
2answers
314 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
1answer
61 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 ...
2
votes
2answers
48 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
votes
1answer
318 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 ...
2
votes
2answers
234 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 ...
2
votes
2answers
162 views

Unions of subspaces

So I need to prove that for the union of $n$ subspaces to be a subspace, each subspace must be a subset of another one of the subspaces. My thought process so far is that I need to prove that it is ...
2
votes
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
3answers
1k views

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 ...
2
votes
1answer
897 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 ...
1
vote
0answers
118 views

Proof that $\text{im}(g)^\top=\ker(f)^\top$ if $\text{im}(f)=\ker(g)$

Let $f:V\rightarrow W$ and $g:W\rightarrow X$ be two linear maps with $\text{im}(f)=\ker(g)$. How do I prove that $\text{im}(g)^\top=\ker(f)^\top$? I am allowed to use the fact that if $f$ is ...
1
vote
3answers
54 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 ...
1
vote
1answer
116 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 ...
1
vote
1answer
84 views

Find a isometry such that the matrix in respect to the canonical basis is:

I need to find a isometry such that the matrix in respect to the canonical basis is: $$\begin{bmatrix}\frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}& 0\\0 & 0 & 1\\x & y & ...
1
vote
3answers
77 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 ${span(k)=S}$ ...