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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2
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2answers
36 views

How to tell if two spherical coordinates lie on the same plane

I have the rho, theta, and phi values of two points, how can one tell that two vectors are normal to the same plane by looking at their spherical coordinates?
3
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5answers
67 views

Proof involving subspaces

I encountered this question in a document I found on a google search, it bugged me because my perception keeps telling me I'm wrong no matter what I do. Let $U$, $W$ and $Z$ be subspaces of a ...
1
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1answer
14 views

showing a basis is suitable for a dual space, from linear forms

Let us define $B = b_1, b_2, b_3$; where $ b_1(f) = f(0)$, $ b_2(f)=−f'(0) $ and $b_3(f) = f''(0) $. Let $E ^∗$ be the dual basis of $E = {1, x, x^2}$. Show that B is a basis of the dual ...
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0answers
20 views

Arrow Space Construction

Is there a paper or book that has rigorously constructed the space of "arrow vectors" and shown that it is a vector space? I'm just wondering how far anyone has followed the heuristic.
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0answers
27 views

Generalization of N-Body Problem

I know the n-body problem has been solved for gravity, but in a purely mathematical sense, has it been solved? Or could it be generalized to any kind of field? Maybe an example will make my question ...
1
vote
1answer
65 views

Steinitz's Lemma - Removing

In the book that I am using, Linear Algebra Done Right, the proof for the Steinitz exchange lemma (which can be found here) left me unconvinced. The proof refers to the linear independence lemma. ...
0
votes
0answers
17 views

Space is direct sum of subspaces - propostion conditions giving me problems

In Sheldon Axler's "Linear Algebra Done Right" - 2$^{\textrm{nd}}$ Edition, on the section for Direct Sums, the following proposition is stated. Following this is the proof of this 'if and only ...
4
votes
1answer
32 views

computation involving exterior $2$-form on $\mathbb{R}^n$

Let $$\theta = \sum_{i=1}^{n-1} x_i \wedge x_{i+1}$$be an exterior $2$-form on $\mathbb{R}^n$, and $A, B \in \mathbb{R}^n$ are vectors$$A = (1, 1, 1, \dots, 1),\text{ }B = (-1, 1, -1, \dots, ...
3
votes
2answers
190 views

Is it possible that there isn't a linear span which precisely spans a vector space?

In the assignment I'm asked to decide whether given: $S = \Bigg \{ \begin{bmatrix}a &b \\ c &d\end{bmatrix} \in M_2(\mathbb{R}) \; | \; ad = 0 \Bigg \},\mathbb{F} = \mathbb{R}$. $S$ is a ...
2
votes
3answers
29 views

What ring-sum of vector spaces can possibly mean?

I'm given this test assignment, and I can't decipher what it says. Would you kindly help me? Here's the assignment itself: Let $U$ and $W$ be sub-spaces of the linear vector space $V$ s.t. $U ...
5
votes
6answers
543 views

Is there such thing as an unnormed vector space?

I learned about Banach spaces a few weeks ago. A Banach space is a complete normed vector space. This of course made me wonder: are there unnormed vector spaces? If there are, can anyone please ...
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vote
2answers
13 views

Vectors: $a = (1,2)$, $ b= (2,-1)$, $c = (-5, 20)$ Find values for $k$ and $l$ for $c = la + kb$

We have these vectors: $a = (1,2)$, $ b= (2,-1)$, $c = (-5, 20)$ and I have to find values for $k$ and $l$ given this: $c = la + kb$ How do I go on about solving this one? Do I have to calculate ...
1
vote
1answer
55 views

Vector Spaces: Tensor Product

Reference Foundation for: Hilbert Spaces: Tensor Product Problem Given a vector spaces $V$ and $W$. Take its algebraic tensor product: $\tau:V\times W\to V\otimes W$ How to prove that the image ...
0
votes
3answers
17 views

To find the two dimensional subspace of $R^{3}$

I am stuck with this question .Kindly help me to get through this Option A is of 1 dimension so it cannot be answer but all other options are looking fine to me , What i am missing ? THANKS
0
votes
1answer
17 views

Null/Col/Row space be a line\plane through the origin?

For a $4\times3$ matrix can the nullspace, the column space and row space all be lines through the origin? For a $2\times4$ matrix can the nullspace, the column space and row space all be planes ...
3
votes
1answer
63 views

Does every isomorphism between $V$ and $V^*$ send some basis to its dual basis?

Suppose that I have a vector space isomorphism $\theta: V \to V^*$ where $V$ is any vector space (probably over $\mathbb{C}$ is required) and $V^*$ is its dual space. Is it always possible to find a ...
13
votes
3answers
276 views

Every invertible linear transformation can be perturbed a bit without destroying invertbility, Neumann series

Let $T: V \to V$ be any linear transformation on a real or complex vector space $V$. Show that there exists $\epsilon_0 > 0$ $($depending on $T$$)$ so that $I + \epsilon T$ is invertible for any ...
0
votes
0answers
16 views

Direct sum of two spaces

Let $\alpha_1=[1,1,0,1]$, $\alpha_2=[1,0,1,1], \alpha_3=[1,1,1,1],\alpha_4=[0,1,1,1]$ be a vectors from $\mathbb{R}^4$ let $U=span(\alpha_1, \alpha_2) \ and \ W=span(\alpha_3, \alpha_4)$ Check that ...
0
votes
0answers
9 views

Reference for work on abelian divisible groups $G$ such that for every $n \in \mathbb N , g \in G , \exists$ unique $x \in G$ such that $g=x^n$

Is there any work or reference in the literature about those abelian divisible groups $G$ such that for every $n \in \mathbb N , g \in G , \exists$ unique $x \in G$ such that $g=x^n$ ; I think then I ...
0
votes
2answers
18 views

Epimorphism of linear transformation

Find values of parameter t for which transformation is epimorphic: $\psi([x_1,x_2,x_3,x_4])=[x_1+x_2+x_3+2x_4,x_1+tx_2+x_3+3x_4,2x_1+x_2+tx_3+3x_4] $ When this transformation is epimorphic i.e. what ...
3
votes
1answer
65 views

Derivability of a function with an infinity of zeroes

Let $F$ be a normed vector space and $a\in F$. Is there a non zero function $f:\mathbb{R}\rightarrow F$, such that $f'(a)=0$ and $f$ is $0$ an infinity of times in any neighborhood of $a$ ? If not, ...
2
votes
1answer
35 views

Faulty proof that $V=U_1 \oplus W$ and $V=U_2 \oplus W$ implies $U_1 = U_2$

The question is as follows: Prove or give a counterexample: if $\ U_1, U_2, W$ are subspaces of $V$ such that $V=U_1 \oplus W$ and $\ V = U_2 \oplus W$, then $\ U_1 = U_2$. I happily ...
0
votes
0answers
15 views

Show that there exists a Hermitian form of signature $(p,q)$.

Let $K = \mathbb{Q}(\sqrt{-2})$ with $V_K = K^n$ considered as a $K$-vector space. Suppose $p, q \in \mathbb{Z}_{>0}$ such that $p + q = n$. Show that for any such $p$ and $q$ there is a Hermitian ...
1
vote
2answers
27 views

Prove vectors create a basis

Let $V$ be a vector space and $U,W,Z$ be it's subspaces where $V=Z \oplus U=Z\oplus W$. We know that $\beta_1,...,\beta_k$ is a basis of $U$ and $\beta_i=\gamma_i+\delta_i$ where $\gamma_i \in Z$ and ...
1
vote
0answers
23 views

Give the following linear transformation find values of parameter

Find values of parameter t for which transformation is epimorphic: $\psi([x_1,x_2,x_3,x_4])=x_1+x_2+x_3+2x_4,x_1+tx_2+x_3+3x_4,2x_1+x_2+tx_3+3x_4 $ When this transformation is epimorphic i.e. what ...
0
votes
0answers
38 views

Can anyone tell whether this vector space question is true or false? [duplicate]

If U and W are subspaces of a finite dimensional vector space V and V=U+W, then dimV≤dimU+dimW. we know that dimV=dim(U+W) and dim(U+W)> dimU+dimW, and therefore dimV>dimU+dimW. I think this is ...
1
vote
1answer
24 views

To find basis of subspace

Let V be subspace of $M_2 (R) $ consisting of all matrices with trace o and such that entries of first row add upto zero.To find basis for this My attempt i have posted here .according to me the ...
2
votes
1answer
46 views

Can anyone check these true and false statements about linear algebra?

For any square matrix $A$, the image of $A^7$ is contained in the image of $A$ I think this question is asking If $A^7x=b$, then $b$ must be in $A$ with some vector $y$ such that $Ay=b$. It Seems ...
1
vote
2answers
26 views

To find dimension of subspace

Let V be subspace of $M_n (R) $ be subspace ofall matrices such that entries in every row add upto zero and entries in every columm also add upto zero .Then i am to find its dimension . I have tried ...
0
votes
1answer
31 views

Orthogonality of remaining non-intersecting basis

Let $A$ and $B$ $\in \mathbb{C}^{4 \times 100}$ be matrices with null spaces $N(A)$ and $N(B)$ respectively. The dimensions of each null space is $96$ and I was able to find that they intersect in ...
2
votes
2answers
21 views

Orthographic projection in euclidean space

Let $E$ be a euclidean space with an inner product given by $$B =\left(\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 2 & -1 \\ 0 & -1 & 2 \end{array} \right) $$ in a basis ...
1
vote
3answers
30 views

Basis of a vector space is a maximal linearly-independent set?

If $V$ is a vector space of finite dimension over $F$, then a basis of $V$ is a maximal, linearly independent set in $V$. Is this conjecture true? If so, how to prove it?
0
votes
0answers
28 views

Understanding change-of-basis and linear operators

First of all , apologies in advance as this isn't so much as a question, but more check of my understanding. Suppose I have an $n$-dimensional vector space $V$ and a given basis $\mathfrak{B}= ...
0
votes
1answer
16 views

Product of $L_2$ norm of vectors

Is the $\sum \Vert b_k\Vert_2^2 \le\ge= \sum \Vert b_k\Vert_2^2 \Vert a_k\Vert_2^2$ ? where $b_k$ is a column vector and $a_k$ is a highly sparse row vector.
0
votes
1answer
37 views

What values must $\alpha$ be so that $F$ is an isomorphic linear transformation? (Bijective)

Let $F:P_2\to P_2$ where $P_2$ is a polynomial vector space with max grade of 2. $$[F]_B= \begin{pmatrix} \alpha & -1 & -1 \\ -6 & \alpha +1 & 0 \\ ...
0
votes
1answer
30 views

How to demonstrate a set is a real vector space (set governed by nonstandard operations)

I am really not that familiar with questions that ask you to work with a operation vector space, even less with the English terms for it. I am... quite lost. How would you prove that it is a real ...
1
vote
2answers
22 views

Proof that the kernel of an endomorphism to the power $n$ is a subset of the kernel of the endomorphism to the power $n+1$

I am expected to know how to prove the following but I can't seem to draw it out. Knowing that V is a Vector Space$$ T:V\to V $$ Prove the following $$ Ker(T^n)\subseteq Ker(T^{n+1}) $$ How ...
1
vote
1answer
15 views

Projection of vectors

Compute $:$ $proj_\vec y (\vec x)$ $\vec{x}_1=\begin{bmatrix} 2 \\ 3 \\ 4 \\ 5 \end{bmatrix}, \vec{y}_2 = \begin{bmatrix} 1 \\ 0 \\ -1 \\ 0 \end{bmatrix}$ Since the projection would be $:$ $(-2/0) * ...
1
vote
2answers
42 views

Will the value of $t$ affect the row, column, and solution spaces?

Consider A = $\begin{bmatrix}4 & 2\\t & 1\\3&t\end{bmatrix}$.Is the column space of $A$ the same for all t$?$Is the row space of $A$ the same for all $t$? Is the solution space of $Ax = ...
-3
votes
2answers
20 views

Problem on CR inequality on finite sum [closed]

Let $f$ be a function from {1,2,3,....,10} to R, s. t. $(\sum_{i=1}^{10}|f(i)|/2^i)^2=(\sum_{i=1}^{10} |f(i)|^2)(\sum_{i=1}^{10}1/4^i)$ mark the correct statement. A. there are uncountably ...
-3
votes
1answer
63 views

Can the nullspace, the column space and the row space all be a line or a plane? [closed]

Can the nullspace, the column space and the row space of a 4x3 matrix all be a line through the origin? Can the nullspace, the column space and the row space of a 2x4 matrix all be a plane through ...
1
vote
3answers
27 views

Showing that the magnitude of the difference of two vectors is larger than the difference of it's vector magnitudes

Long title. I have to prove (the problem itself suggests using Pythagorean theorem) the following inequality: $$\|u\|-\|v\| \le \|u-v\| $$ Vector magnitudes... How do you prove this in an ...
2
votes
2answers
31 views

The geometric meaning of a line plus a vector

Lets say we have $$ E = \{k(1,2,3)' + (2,9,-1)'\} \;\mathrm{with}\; k \in \mathbb{R} $$ we know that $k(1,2,3)$ spans a line in three dimensions, but what does the shape of $E$ look like. I think it ...
2
votes
3answers
107 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
2answers
63 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 ...
1
vote
3answers
61 views

Question on the definition of vector spaces.

My question is perhaps useless, but I want to shed some clarity on this matter. I'm bothered by people that say a vector space is a "bunch of vectors". Or that a vector space "consists of ...
3
votes
1answer
48 views

Vector Spaces and Groups

I've just completed a course in linear algebra. I'm a physics undergraduate and I don't plan on taking an abstract algebra course. That said, I've been reading a little bit about it. As I understand ...
3
votes
2answers
30 views

Norm of linear transformation: why restrict ourselves to $\|x\|\leq 1$?

If $f$ is linear transformation from a normed linear space $X$ into a normed linear space $Y$, and define its norm by $$\|f\|=\sup\{\|f(x)\|: x\in X, \ \|x\|\leq 1\}$$ My question is: why restrict ...
3
votes
2answers
42 views

Orthonormal basis . Can I have more than one basis for the subspace?

Required to find an orthonormal basis for the following subspace of R4 I know that to find the othonormal basis, it is required that i find the basis for the subspace, then I use Gram Schmidt ...
0
votes
1answer
23 views

Gram-Schmidt process in function subspace

I have a function space $\mathcal {F}([-1,1],\mathbb R)$ and the subspace $\mathcal{P_2}:=$ $(x\mapsto a_o+a_1x+a_2x^2| a_0,a_1,a_2 \in \mathbb R )$ for all polynomials with degree $\le2$. In this ...