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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3answers
129 views

$I+A^*A$ is non-singular whenever $A$ is a square matrix with complex entries? [closed]

Let $A$ be a square matrix with complex entries , then is it true that $I+A^*A$ is non-singular ? where $A^*$ denotes the conjugate transpose of $A$ http://en.wikipedia.org/wiki/Conjugate_transpose ...
0
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1answer
91 views

Finding loci of possible points satisfying vector simultaneous equations

I recently had an exam and a question came up which I was only partially able to answer. The question was the following: Let $\vec{a}$, $\vec{b}$ and $\vec{c}$ be constant vectors in ...
0
votes
1answer
148 views

inner product space , dual space, proof about isomorphism

Let $V$ be a vector space (not necessary being finite dimensional) and let $U,W$ be subspaces of $V$ such that $V = U\oplus W$. Prove that $V^\ast/(W^0)$ is isomorphic to $W^\ast$. Notation and ...
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1answer
76 views

V = U⊕W then Prove that (V/W)* is isomorphic to W^0

Let $V$ be a vector space (not necessary being finite dimensional) and let $U$, $W$ be subspaces of $V$ such that $V = U\oplus W$. Prove that $(V/W)^*$ is isomorphic to $W^0$. note: (V/W)* is the ...
0
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3answers
92 views

Linear Algebra: Matrix Spanning/Consistency Question

1) If there are $5$ vectors found in $\mathbb{R}^7$ will these vectors Span $\mathbb{R}^7$? Please explain. 2) Give an example of a $3$ by $5$ matrix for which all systems, $Ax=b$ for any $b$ in ...
0
votes
4answers
74 views

A vector should more be thought an identity of an entity in space rathar than magnitude + direction?

Can I say that vector is more like a "unique identity" of an entity in space rather than calling it an entity with magnitude and direction ? For example a line. A vector $(10,10,0)$ is the identity ...
0
votes
0answers
101 views

Finding a Vector not in a Collection of Proper Subspaces

Let's take $K$ to be an infinite field and $V$ a vector space over it. Allowing $U_1,...U_l$ to be $l$ proper subspaces of $V$ (none the same), and further assuming that for all $i$, $U_i$ is not in ...
0
votes
1answer
307 views

How is it possible for the inverse function of a linear-continuous-bijective function to be not continuous?

If $E$ and $F$ are two normed vector spaces, $f:E\rightarrow F$ is a linear-continuous-bijective function. Then naturally I would think that $f^{-1}$ is also linear-continuous-bijective. But the ...
0
votes
1answer
274 views

Visualization of 2-dimensional function spaces

As a follow-up question to what is the norm measuring in function spaces I just had an idea: How about visualizing function spaces as normal planes. What I have in mind is to have an orthogonal ...
-1
votes
2answers
109 views

Definition of dimension

Let us consider Euclidean space $\mathbb{R}^n$. We say it is $n$-dimensional because each vector in it is an $n$-tuple $(x_1,...,x_n)$. However, it is possible to represent this exact same space using ...
9
votes
2answers
225 views

If $V \times W$ with the product norm is complete, must $V$ and $W$ be complete?

Let $V,W$ be two normed vector spaces (over a field $K$). Then their product $V \times W$ with the norm $\|(x,y)\| = \|x\|_V + \|y\|_W$ is a normed space. Using this norm it's easy to show that if ...
8
votes
2answers
224 views

Can a basis for a vector space $V$ can be restricted to a basis for any subspace $W$?

I don't understand why this statement is wrong: $V$ is a vector space, and $W$ is a subspace of $V$. $K$ is a basis of $V$. We can manage to find a subset of $K$ that will be a basis of $W$. ...
7
votes
3answers
403 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 ...
6
votes
1answer
78 views

Invariant Subspace of Two Operators [duplicate]

Let $S$, $T$ be linear operators on a finite-dimensional vector space $V$ over $\mathbb{C}$. Suppose $$S^2 = T^2 = I.$$ Show that there exists either a $1$-dimensional or $2$-dimensional ...
6
votes
1answer
174 views

How many parameters are required to specify a linear subspace?

A problem in Peter Lax's Linear Algebra involves looking at the family of $n\times n$ self-adjoint complex matrices and asking: on how many real parameters does the choice of such a matrix depend? ...
6
votes
3answers
336 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
1answer
146 views

Product of binomial coefficient as a basis

I am stuck with the following problem. Every polynomial of degree $d$ can be expressed as $$ p(x) = p_d \binom{x}{d}+ p_{d-1}\binom{x}{d-1} + \cdots + p_0 \binom{x}{0} $$ What is the ...
4
votes
1answer
98 views

Which rings containing the complex field are, as vector spaces over that field, isomorphic to $\mathbb{C}^2$?

Which rings $R$ containing (as a subring) the complex field $\mathbb{C}$ are, as a vector space over that field, isomorphic to $\mathbb{C}^2$? In other words: what are the two-dimensional unital ...
4
votes
1answer
61 views

Project a vector onto the intersection of surfaces

I want to project a vector $\vec v$ onto a surface $S$ defined as the intersection of other surfaces. For example, in 5-dimension I have the surface $S(x_1,x_2,x_3,x_4,x_5)=c$, defined by the ...
4
votes
2answers
81 views

Show ker($\alpha$)=ker($\alpha$)^2 iff ker($\alpha$) and im($\alpha$) are disjoint

Let $V$ be a vector space over a field $F$ and let $\alpha$ be an element of $\operatorname{End}(V)$. Show $\ker(\alpha)=ker(\alpha^2)$ iff $\ker(\alpha)$ and $\operatorname{im}(\alpha)$ are disjoint. ...
4
votes
1answer
49 views

Projection onto space in $L^2([0,1])$ gives shorter length

Let $f_1,f_2,\ldots,f_n\in L^2([0,1])$, and let $V$ denote their span. Let $P:L^2([0,1])\rightarrow V$ be the projection onto $V$. Let $g\in L^2([0,1])$. Suppose also that $g\in L^p([0,1])$ for some ...
4
votes
2answers
193 views

Show that $F+G$ is closed when $G$ a closed subspace of normed space $E$ and $F$ a finite dimensional subspace of $E$.

Question: Let $E$ be a normed space. Let $G$ be a closed subspace of $E$ and let $F$ be a finite dimensional subspace of $E$. Show that $F+G$ is a subspace of $E$ and is closed. I'm having trouble in ...
4
votes
1answer
211 views

Normal $T\in B(H)$ has a nontrivial invariant subspace

I am wondering if the following is true: Every normal $T\in B(H)$ has a nontrivial invariant subspace if $\dim(H)>1$?
4
votes
1answer
414 views

Algorithm to find the basis of intersection of subspaces without gaussian elimination.

Is there an algorithm to find the basis of intersection of subspaces $A_1$ and $A_2$, if we have the bases of subspaces $A_1$ and $A_2$, without using Gaussian elimination? Thanks.
4
votes
1answer
523 views

When are two diagonal matrices congruent?

This is probably a question that does not admit a simple answer. However, I'd like to know whether there exist criteria that determine when two diagonal matrices are congruent. I have the suspicion ...
4
votes
3answers
543 views

Does $\det(A) \neq 0$ (where A is the coefficient matrix) $\rightarrow$ a basis in vector spaces other than $R^{n}$?

I know that for a set of vectors $\{ v_{1}, v_{2}, \ldots , v_{n} \} \in \mathbb{R}^{n}$ we can show that the vectors form a basis in $\mathbb{R}^{n}$ if we show that the coefficient matrix $A$ has ...
4
votes
1answer
392 views

Prove a basic fact on a linear combination of vectors

Let $x_i (i=1,...,n, n>d)$ be a unit vector in $R^d$. $c_i>0$ is a positive real scalar. How to prove the following fact? Fact: There exist some vectors $x_i$ such that $\sum_{i=1}^n c_i ...
4
votes
2answers
261 views

Dot product of two vectors

How does one show that the dot product of two vectors is A · B = |A| * |B| * cos(Θ) ?
4
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2answers
485 views

Is $(\mathbf{V} \cap \mathbf{W})^{\bot}=(\mathbf{V}^{\bot} \cap \mathbf{W}^{\bot})$?

Is $(\mathbf{V} \cap \mathbf{W})^{\bot}=(\mathbf{V}^{\bot} \cap \mathbf{W}^{\bot})$? I tried element-chasing, but I am getting confused when trying to determine mutual containment.
3
votes
2answers
88 views

Prove that, if $\{u,v,w\}$ is a basis for a vector space $V$, then so is $\{u+v, v+w, u+v+w\}$.

I'm trying to prove the following statement: In a vector space $V$ over a field $\mathbb{F}$, if $\{u,v,w\}$ is a basis for $V$, then $\{u+v, u+v+w, v+w\}$ is also a basis. $$\underline{\text{My ...
3
votes
3answers
269 views

Vector space of continuous [0,1] -> [0,1] over R

Is there a way to define a vector space of continuous functions of type $[0,1] \to [0,1]$ over $\mathbb R$? If no, how to prove that such a vector space does not exist?
3
votes
1answer
116 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
1answer
662 views

Prove $\mathbb R$ vector space over $\mathbb Q$

I am proving that $\mathbb R$ is a vector space over $\mathbb Q$. So far, I have stated that vector addition and scalar multiplication trivially hold in $\mathbb R$. I then showed that $(\mathbb R, ...
3
votes
4answers
314 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 ...
3
votes
2answers
2k views

Relation between Interior Product, Inner Product, Exterior Product, Outer Product..

Following my previous question Relation between cross-product and outer product where I learnt that the Exterior Product generalises the Cross Product whereas the Inner Product generalises the Dot ...
3
votes
1answer
180 views

Define two differents vector space structures over a field on an abelian group

Exercise 3 from Roman's book "Advanced Linear Algebra". The author asks us to "find an abelian group $V$ and a field $\mathbb{F}$ for which $V$ is a vector space over $\mathbb{F}$ in at least two ...
3
votes
1answer
749 views

How many different basis' exist for an n-dimensional vector space in mod 2?

Imagine a 2-dimensional vector space in $\mathbb{Z} /2 \mathbb{Z}$. The only possible basis' are $$ \left(\begin{array}{c} 1\\ 0 \end{array}\right), \left(\begin{array}{c} 0\\ 1 ...
3
votes
2answers
229 views

How to prove the inequality $\Theta(x,y)\le \Theta(x,z)+\Theta(z,y)$?

Let $x, y$ be two complex vectors, $$\cos\Theta(x,y):=\operatorname{Re} \frac{y^*x}{\|x\|\|y\|} .$$ Then I want to prove that $$\Theta(x,y)\le \Theta(x,z)+\Theta(z,y) .$$
2
votes
2answers
79 views

Prove that a subspace of dimension $n$ of a vector space of dimension $n$ is the whole space.

Maybe this is a stupid question. I was brought to this from the observation that an infinite dimensional vector space can have proper subspace that have the same dimension of the whole space. But, ...
2
votes
2answers
41 views

How to figure the size of the following vector set?

Let $V$ be the set of all vectors in $\mathbb R^n$ with entries $±1$. What is the size of this vector set? I know the answer is $2^n$ but I cannot prove why. I feel like this has something to do ...
2
votes
2answers
854 views

Linear Algebra, Vector Space: how to find intersection of two subspaces ?

$${ W = Sp\{{(1,3,4),(2,5,1)\}}\\ U = Sp\{{(1,1,2),(2,2,1)}} \}$$ Find a span $${U \bigcap W}$$ First time using Math latex, pretty hard.
2
votes
3answers
458 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
1answer
118 views

Linear Transformation defined by a Matrix and Invariant Subspaces

I got stuck solving this problem: Let $T:\mathbb{R}^3\to \mathbb{R}^3$ be the linear transformation defined by the matrix A in the standard basis of $\mathbb{R}^3$, $E=\{e_1,e_2,e_3\}$ ...
2
votes
4answers
418 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!
2
votes
4answers
210 views

Intuition behind definition of transpose map

The linear map $T^t:V' \to V'$ (where $V'$ is the set of all linear maps from $V$ to its scalar field $\Bbb F$) is defined by : $$T^t(f)(v)=f(T(v))$$ This looks like some "commutative" definition ...
2
votes
1answer
471 views

Calculate angle of triangle

I need to calculate the angle between two sides, I have the length of A & B sides, but don't know how to find the angle... Both sides are the same length. I can get the start and end vectors of ...
2
votes
1answer
2k views

Relationship between covariant/contravariant basis vectors

I'm starting to learn some of the basics of covariant and contravariant vectors. I'm a little confused about the difference between a covariant and a contravariant basis vector. I know that the ...
2
votes
2answers
94 views

How to show $X=\{A\in\mathcal{L}(\mathbb R^m, \mathbb R^n); \textrm{Ker}(A)=\{0\}\}$ is open in $\mathcal{L}(\mathbb R^m, \mathbb R^n)$?

How to show $X=\{A\in\mathcal{L}(\mathbb R^m, \mathbb R^n); \textrm{Ker}(A)=\{0\}\}$ is open in $\mathcal{L}(\mathbb R^m, \mathbb R^n)$? Here $\mathcal{L}(\mathbb R^m, \mathbb R^n)$ is the set of ...
2
votes
4answers
95 views

Should I use sets or tuples when dealing with linear dependence?

Let set of vectors $\{x,y,z\}$ be linearly independent. Then would $\{x,y,z,x\}=\{x,y,z\}$ be linearly dependent, also? If so, that seems like a problem (since $\alpha x+\beta y+\gamma ...
2
votes
1answer
175 views

Counting automorphisms

How does one count the number of automorphisms of a vector space? If a vector space over $\mathbb F_p$ has $n$ ordered bases how many are there? I think I should be considering the mappings of a set ...