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

4
votes
6answers
1k views

“Vectors aren't really numbers” - how sound is that statement?

Since I first learned about vectors, I noticed something interesting: almost any numeric formula can be replaced by a vectorial formula by just replacing addition, multiplication, etc., with their ...
0
votes
1answer
25 views

Checking vectors for subspaces in $\mathbb{R}^3$ space.

How to check if these sets are subspaces in $\mathbb{R}^3$ ? i know the three condtions but how to check those conditions with some solvings? Thanks in advance...... $$U_1 = \{(x,y,xy)\mid x,y ∈ ...
0
votes
2answers
36 views

Proof that $V^*$ is isomorphic to $V$.

In my notes for a linear algebra course there is proof that $V^*$ is isomorphic to $V$. However I am unclear on a few of the steps. We begin by choosing a basis $B = \{v_1,...,v_n\}$ for $V$. We now ...
1
vote
1answer
31 views

Clarification about Quotient Spaces

The question given to me is: Does there exist a vector space $V$ which has a nonzero subspace $U$ such that $V /U \cong V$ ? Provide an example or a proof that no such $V /U$ exists. Intuitively, I ...
0
votes
3answers
22 views

How do I find a dual basis given the following basis?

$V = \Bbb{R}^3$ and has basis $\mathcal{B} = \{\vec{e_1}-\vec{e_2},\vec{e_1}+\vec{e_2},\vec{e_3}\}$ How do I find the dual basis? This is not homework, but an example that I am struggling to grasp. ...
0
votes
1answer
27 views

base of the vectorspace $\Bbb{Q}(a)$ over $\Bbb{Q}$

i have to find the base of the vectorspace $\Bbb{Q}(\alpha)$ over $\Bbb{Q}$ with $\alpha = \sqrt{1+\sqrt{3}}$. i have found the minimal polynom of $\alpha$ over $\Bbb{Q}$: $f(x)=x^{4}-2x^{2}-2$. The ...
3
votes
1answer
24 views

Vector calculus problem, constant speed, counterclockwise or clockwise.

I'm stuck on how to do this problem: $\displaystyle \vec{r}(t)=(\cos t)\,\vec{i}+(\sin t)\,\vec{j}, \qquad t \geq 0.$. Does the particle have constant speed? (yes or no) For this one I was thinking ...
0
votes
2answers
33 views

Show that V is a vector space over the set of real numbers when V is the set of all real 3x3 matrices

Wondering how one would go on about this. V is the set of all real 3 × 3 matrices. How can it be shown that V is a vector space over the set of real numbers and what would be the dimension of and ...
0
votes
2answers
19 views

Orthogonal Projection in subspace

Consider the vector space $\mathbb{R}^n$ with usual inner product $<.,.>$. Take $Y\in \mathbb{R}^n$ and $X \in \mathbb{R}^n$ such that $Y=[y_1,y_2,..y_n]^t$ and $X=[1,1,....1]^t$ ...
2
votes
2answers
40 views

What is the difference between $n$-tuples, $m \times 1$ and $1 \times n$ matrices?

Isn't the tuple different structure from $m \times 1$ or $1 \times n$ matrix? Why can you mix them?
-1
votes
0answers
14 views

Find a vector equation for the tangent line to the curve r⃗(t)=(3cos(2t))i⃗+(3sin(2t))j⃗+(sin(3t))k⃗ at t=0. [closed]

So I was thinking of finding the derivative of this vector, and I got <-6sin(2t), 6cos(2t),3cos(3t)>. I plugged in 0 for t and I got <0,6,3>, but that is not the correct answer since i have to ...
2
votes
1answer
21 views

What is the fraction of volume of unit hypersphere centered at one of the vertices of hypercube to that of hypercube?

consider a hyper-cube of n-dimension having a length of "r" units across each dimension. If a unit n-dimensional sphere is present at one of the vertices of the hyper-cube. what fraction of volume of ...
0
votes
1answer
25 views

How would I find the acceleration of this vector?

So I found the velocity already (which my homework says it's correct). The velocity is i+2tj+4k. I know the acceleration is the derivative of the velocity. I found it to be 1+2j+4 , but when I enter ...
0
votes
1answer
53 views

Find a function $f(x)$ so that the graph of $y=f(x)$ is the path of the particle.

The equation $r(t) = \frac{t}{t+4} \vec i + \frac{4}{t} \vec j$ gives the position of a particle in the $xy$-plane at time $t$. Find a function $f(x)$ so that the graph of $y=f(x)$ is the path of ...
1
vote
1answer
31 views

Question on meaning of notation

What does the following mean in context to Differential Geometry? The book I am reading uses it without explanation. $${\left. {Df\,} \right|_u}(V)$$
0
votes
1answer
33 views

understand an answer to linear span of polynomal subspace exercise

i am looking at an answer to an exercise who asks to find a linear span for, and I don't really understand the solution $$p(x) = ax^3 + bx ^2 + cx + d$$ and this is the solution i see $$ p(x) \in M ...
1
vote
1answer
31 views

find equality between linear spans

$$U = Sp\{(2,5,-4,-10), (1,1,1,1),(1,0,3,5), (0,2,-4,-8)\}$$ $$ W = Sp\{(1,-2,7,13), (3,1,7,11), (2,1,4,6) \}$$ two questions: prove that $U = W$ find the values of the $a \in \mathbb{R}$ where the ...
14
votes
4answers
718 views

Vector Spaces: Redundant Axiom?

Question Why are the axioms for vector space independent? More precisely $1x=x$ seems redundant... (I take the axioms from: Wikipedia) Explanation One has for zero vector: ...
0
votes
2answers
19 views

linear span of subspace

we have the following subspace over $\mathbb{R}$ $$M = \{ A \in M^{{n\times n}} | A = -\overline{A} \}$$ I found that it is a subspae and now I need to find the linear span of it. how can I calculate ...
0
votes
2answers
17 views

Finding if a group of polynom field is a vector space

we have the following group $$P = \{p(x) \in \mathbb{R}^4[x] | p (x) = p (1-x)\} $$ and I need to find out if this is group is vector space or not. and I am not even sure what could $P$ contain. I ...
0
votes
1answer
23 views

Finding if a group is a vector space

$\mathbb{C}^2$ is a group over field $\mathbb{C}$, with the following actions: addition is similar to the regular addition. multiplication is defined by: for every $(z,w) \in \mathbb{C}^2$ and every ...
1
vote
0answers
17 views

Proving vectors as a basis in $E^{m}$

Show that if the vectors $a_{1}$, $a_2$, $\cdots$, $a_m$, are a basis in $E^{m}$, the vectors $a_{1}$, $a_2$, $\cdots$, $a_{p-1}$, $a_{q}, a_{p+1}, \cdots,a_{m}$, also are a basis if and only if ...
0
votes
0answers
103 views

Linear Algebra- Subspace Question

"Is the set of all polynomials over the real numbers of degree exactly 2 a subspace of P∞(R)?" So apparently the answer to this is 'no'. Can anyone explain why?
1
vote
2answers
89 views

How is the vector space of abstract “tuples” isomorphic to vector space of $n \times 1$ or $1 \times n $ matrices?

I read that the vector space of abstract "tuples" is isomorphic to vector space of $n \times 1$ or $1 \times n $ matrices. Where can I find a good explanation of this or can someone explain it here?
0
votes
0answers
13 views

How to find the value of standard coordinate frame in a new coordinate frame?

I have a custom coordinate frame which has T as a point and A, B, C are three orthogonally normalized vectors whose coordinates are T = [Xt Yt Zt], A = [Xa Ya Za], B = [Xb, Yb, Zb] and C = ...
1
vote
1answer
30 views

finding the matrix representation of a linear transformation

i was having hard time solving this one, any help will be greatly appreciated: given the following linear transformation: $$ T(X) := BX^t-XB^t $$ $$ B= \left(\begin{matrix} 0 & 1 \\ ...
3
votes
1answer
25 views

Establish natural isomorphism: $\mathcal{B}(E,F;G) \cong \mathcal{L}(E;\mathcal{L}(F;G))$

Where $\mathcal{B}(E,F;G)$ is the space of bilinear functions from vector spaces $E \times F \rightarrow G$ and $\mathcal{L}(E;\mathcal{L}(F;G))$ is the space of linear functions from $E \rightarrow ...
0
votes
1answer
20 views

solving a set of vector equations

Let's have six vectors $\boldsymbol p$, $\boldsymbol r$, $\boldsymbol s$, $\boldsymbol t$, $\boldsymbol u$, $\boldsymbol v$ from $\mathbb{R}^N$. We are given the following two vector equations: $$a_1 ...
1
vote
2answers
48 views

How is it distinguished in matrix multiplication which is the vector and which is the matrix representing a linear transformation?

The terminology that is used everywhere when applying a matrix to a "vector" is considered is this: the matrix represents a linear transformation and there is a row or column vector. But a matrix can ...
-2
votes
1answer
87 views

Consider I'm a 10 year old kid, explain what “linearly independent” and “basis” means [closed]

As the question states. Consider I am a child, explain what those concepts mean.
1
vote
2answers
27 views

Find a vector equation for the line through the point $(3,-8,-8)$ perpendicular to these vectors

Find a vector equation for the line through the point $(3,-8,-8)$ perpendicular to these vectors $u=\langle 2,2,-1\rangle$ and $v=\langle -9,-8,-3\rangle$. I'm fairly new to vector equations. ...
3
votes
0answers
14 views

Find a matrix to represent the mapping of a factor module

I have a problem from my past paper I can't figure the logic to, even after seeing the answers. The question goes 【Q】Let $V=\mathbb{R}[X]_{<4}$ be the vector space of real polynomials of degree ...
-2
votes
1answer
28 views

Have a question about Linear Transformations

Explain why there cannot be a linear transformation T: $R^2$ --> $R^2$ for which T(1,1)=(2,3) and T(3,3)=(1,4). I have no clue how to start this problem. Wouldn't ...
2
votes
1answer
35 views

Theorem 3.1 in Erwin Kreyszig's “Introductory Functional Analysis With Applications”: Is the notion of convex set valid in complex vector spaces?

Is the notion of convex sets valid and meaningful for complex vector spaces? Or, do we need to restrict ourselves to real vector spaces and normed spaces when we talk about convex sets? The ...
0
votes
0answers
17 views

Let L be the set of solutions (x,y,z) in R^3 to the equation x - 5y - 2z = 1. Find a vector u ϵ R^3 and a subspace W of R^3 such that L = u + W

My attempt so far: 1) Expressing L in parametric vector form: L = y(5,1,0) + z(2,0,1) + (1,0,0) 2) L = u + W <=> W = L-u 3) W is a subspace if it is closed under vector addition, closed under ...
2
votes
1answer
33 views

Operator that mantains unit vector

Let $\hat u\in\mathbb R^m$ and $\hat v\in\mathbb R^n$ (with $m \neq n$) represent unit vectors in different vector spaces, and let $B$ be a matrix such that: $$ B\cdot\hat u=\hat v $$ What kind of ...
-1
votes
2answers
28 views

Vector Inequality.

Given that $|a|=|b|=1$ and $c=a\times b$ then maximum value of: $$|(a+2b+3c).((2a+3b+c)\times(3a+b+2c))|$$ And I know this is the box product: $$[(a+2b+3c),(2a+3b+c),(3a+b+2c)]$$ And hence the volume ...
0
votes
0answers
24 views

Upperbound for a linear algebraic ratio?

Consider ($n\times 1$)-column vector $\mathbf{p} = (p_i)_{i=1}^n$ with $p_i > 0$ and a symmetric ($n\times n$)-matrix $\mathbf{A} = [a_{ij}]$ with $a_{ii} = 0$ and $a_{ij} \in [0,1]$ for $i \neq ...
0
votes
2answers
18 views

Surd-like trinomials form a field

This is a problem from Artin's book "Algebra". In the fifth miscellaneous problem of the chapter "Vector spaces", he has asked to prove that: If $\alpha$ is a cube root of $2$, then the real numbers ...
0
votes
1answer
39 views

The property of closed subspace

We know that a set is closed if and only if every convergent sequence with elements in the set has a limit point in the set. I am reading a paper, and the paper claims that the following is due to S ...
0
votes
3answers
32 views

Need help finding the projection of a vector onto a subspace.

(a) Find the projection of the vector $\vec b=(2,1,0,1)$ onto the subspace $V$ consisting of all vectors of the form $(x_1,x_2,x_3,x_4)$ such that $x_1+x_2+x_3+x_4=0$. (b) What is the distance from ...
8
votes
5answers
591 views

Can the integers be made into a vector space over any Finite Field?

Given a Finite Field $F$, can the the abelian group $\mathbb Z$ be made into a vector space over $F$ without changing the additive structure of $\mathbb Z$? This seems like it shouldn't be ...
2
votes
1answer
20 views

Proving surjectivity and injectivity of two transformations, knowing the rank of their composition.

I have got another question concerning linear algebra. The excercise is: Let ...
0
votes
2answers
38 views

Proving that there is a unique linear map such that $T(u_i)=v_i$.

I have a problem with understanding of a rather simple concept in linear algebra. I have seen in a book, a following question: Suppose $U,V$ are vector spaces over $K$ and $u_1,\dots,u_n$ is a ...
0
votes
0answers
20 views

Necessity of continuity in Topological Vector Space

In the notion of a topological vector space, we define such as a vector space $X$ (over a field $\mathbb{K}$) with topology $\mathscr{T}$ such that $$\iota_+: (X,\mathscr{T}) \times (X, \mathscr{T}) ...
2
votes
2answers
34 views

Does a direct sum decomposition of an infinite-dimensional vector space require Zorn's lemma?

Let $V$ be an infinite-dimensional vector space and $V'\subset V$ a subspace. Does it require Zorn's lemma to write $V=V'\oplus V''$ for some other subspace $V''\subset V$?
0
votes
1answer
33 views

Proving that a set is a subspace of a given space.

I have encountered this question and I am wondering whether my thinking is correct. We have vectors $u_1, u_2, \dots, u_n$ as elements of the space $V$. We have also got a set $U$ is the set of all ...
2
votes
1answer
22 views

Prove relationship regarding the scalar product

For 2 vectors $a,b$ $\in \mathbb{R}^n$ and all entries in the vectors are $\geq 1$ is the following relationship true ? : $\langle a,b \rangle$ $ \leq$ $0.5 \langle a,a \rangle + 0.5 \langle b,b ...
0
votes
2answers
18 views

Determining if vector space holds

Let A be a particular vector in $\Bbb R$2x2. Determine whether the following is a subspace of $\Bbb R$2x2: S = {B ∈ $\Bbb R$2x2 | AB + B = O} I have two ideas on how to approach this for scalar ...
1
vote
1answer
24 views

Factor Modules/Vector Spaces and its basis with canonical mappings

I'm having trouble with factor modules now. Well, specifically the following question from a past paper. Q. T=$R^2$ i.e. the real plane, and define $f:T$->$T$ with respect to the standard basis which ...