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
213 views

Algebraic complements in vector space of functions without the axiom of choice

The axiom of choice is equivalent to the statement that every subspace $U$ of every vector space $V$ has an algebraic complement, i.e. another subspace $W$ that has a trivial intersection with the ...
2
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0answers
174 views

Horizontal and vertical tangent space of Orthogonal group

We know for the orthogonal group n-by-n orthogonal matrices, the tangents are given by $X^T\Delta + \Delta^TX = 0$ where $\Delta$ is the tangent. Now I was reading about the vertical and horizontal ...
2
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2answers
313 views

short exact sequence of holomorphic vector bundles splits but not holomorphically, only $C^{\infty}$

If there is a short exact sequence of holomorphic vector bundles, $$0 \overset{a_1}{\to} W \overset{a_2}{\to} V \overset{a_3}{\to} F \overset{a_4}{\to} 0,$$ then one can expect a $C^{\infty}$ ...
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2answers
93 views

What is the product and coproduct of Morphism category(Arrow category)?

Given category C, Its morphism category D means a category that has 1) "morphisms of C" as its objects 2) "pair (f,g) s.t. the diagram(square) commutes" as its morphisms The above definition is ...
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3answers
103 views

What's special about the first vector

My linear algebra notes state the following lemma: If $(v_1, ...,v_m)$ is linearly dependent in $V$ and $v_1 \neq 0$ then there exists $j \in \{2,...,m\}$ such that $v_j \in span(v_1,...,v_{j-1})$ ...
1
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4answers
191 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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3answers
468 views

Induced Exact Sequence of Dual Spaces

So given a short exact sequence of vector spaces $$0\longrightarrow U\longrightarrow V \longrightarrow W\longrightarrow 0$$ With linear transformations $S$ and $T$ from left to right in the ...
1
vote
1answer
163 views

Dimensions of vector subspaces

Given a bilinear map $B:X\times Y\to F$ where $X,Y$ are vector spaces and given $S\leq X$, why is $\dim S+\dim \operatorname{ann}(S)=\dim Y$ where $\operatorname{ann}(S)$ is the annihilator of $S$ ...
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3answers
43 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
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4answers
69 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 ...
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1answer
80 views

Proof of $(u.v)=(u_{x}.v_{x}+u_{y}.v_{y}+u_{z}.v_{z})$, assuming $(u.v)=|u||v|cos\theta$.

I would really love any sort of proof of this. I have a very elementary geometric proof for $R^{2}$. That's mainly because I can easily represent $cos\theta$ in the form of ...
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1answer
106 views

Problem related to differential of a map

I dont understand how to solve this problem. Please can you explain the solution clearly? I want to learn how to solve such problems. Thank you
0
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1answer
197 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
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0answers
201 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 ...
9
votes
2answers
195 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
177 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
226 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
122 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
180 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
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1answer
742 views

Does the multiplicative identity have to be 1?

I am just starting out with vector spaces and I am having a hard time understanding them. One of the requirements states that $1\mathbf{v}=\mathbf{v}$ where $1$ is the multiplicative identity. Does 1 ...
4
votes
1answer
47 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
3answers
153 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
155 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
320 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
412 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
407 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
283 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
226 views

Dot product of two vectors

How does one show that the dot product of two vectors is A · B = |A| * |B| * cos(Θ) ?
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2answers
425 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
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1answer
90 views

If the expectation $\langle v,Mv \rangle$ of an operator is $0$ for all $v$ is the operator $0$?

I ran into this a while back and convinced my self that it was true for all finite dimensional vector spaces with complex coefficients. My question is to what extent could I trust this result in the ...
3
votes
3answers
256 views

Writing u as a linear combination of the vectors in S.

Write vector u = $$\left[\begin{array}{ccc|c}2 \\10 \\1\end{array}\right]$$ as a linear combination of the vectors in S. Use elementary row operations on an augmented matrix to find the necessary ...
3
votes
4answers
161 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
284 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
408 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
2answers
1k 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
2answers
226 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
1answer
42 views

What is the difference between Cartesian and Tensor product of two vector spaces

In particular, how is it that dimension of Cartesian product is a sum of dimensions of underlying vector spaces, while Tensor product, often defined as a quotient of Cartesian product, has dimension ...
2
votes
1answer
85 views

Problem with Alternate Solution - Equation of plane through point and containing intersection line of two planes [Stewart P$803\, 12.5.37$]

$37.$ Find an equation of the plane that passes through the point $(1, -2, 1)$ and contains the line of intersection of the planes $x + y - z = 2$ and $2x - y + 3z = 1$. $\bbox[3px,border:2px solid ...
2
votes
1answer
66 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
136 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
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1answer
239 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
2answers
91 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
87 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
1k views

Equation for non-orthogonal projection of a point onto two vectors representing the isometric axis?

Suppose I have two vectors that are not orthogonal (let's say, an isometric grid) representing the new axis. Suppose I want to project a point onto these two vectors, how would I do it? Dot product ...
2
votes
1answer
144 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 ...
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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 = ...
2
votes
1answer
164 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 ...
2
votes
0answers
202 views

Direct sum and direct product of vector spaces [duplicate]

Possible Duplicate: The direct sum $\oplus$ versus the cartesian product $\times$ (Definition) I was wondering how their definitions are different? Are they both the cartesian product with ...
2
votes
2answers
6k views

plane determined by 2 vectors

i have 2 perpendicular vectors in space . How can i determine the plane determined by the 2 vectors? Regards, Alexandru Badescu
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3answers
1k 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 ...