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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3answers
18 views

How much should I scale $dx$ and $dy$ individually to get a vector of required magnitude

I have a $dx$ and a $dy$ and I need to create a vector of magnitude $35.5$ in that $(dx, dy)$ direction. How much should I scale $dx$ and $dy$?
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1answer
32 views

Determining a basis for a space of polynomials

Determine a basis from the following set of second degree polynomials. Does this basis span the space of the second degree polynomials? What is the dimension of the (sub)space that it spans? ...
-1
votes
2answers
30 views

Whether a set of vectors span a subspace that includes a given vector

Do the vectors $(0, 1, 2), (1, 2, 1), ( -1, 2, 4)$ a) span $\mathbb R^{3}$ b) span a subspace that includes $w = (-2, 2, 10)$ I know they don't span $\mathbb R^3$ since they are ...
1
vote
3answers
27 views

Property of eigenvectors in linear mapping

Let $V$ be a bector space over a filed $\mathbb{F}$, and let $L:V\rightarrow V$ be a linear mapping. Let $U$ be a subspace of $V$ such that $L(U)\subset U$ Suppose that $u$ and $v$ are eigenvectors ...
3
votes
1answer
27 views

Prove dimension of sum of two subspaces

Let $U$ and $W$ be subspaces of $\mathbb{R^n}$ where $\dim(U)=n-1$, $\dim(W)=n-3$ and $n\geq 3$ Prove that $\dim(U\cap W)\geq n-3$ I used the property that both $U$ and $W$ are subspaces of ...
2
votes
2answers
38 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?
4
votes
5answers
174 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
vote
1answer
17 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 ...
0
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0answers
24 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.
1
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1answer
66 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
35 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
195 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
31 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
545 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 ...
1
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
57 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
70 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
278 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
66 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
36 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
16 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
28 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
16 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
43 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 ...