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

1
vote
1answer
32 views

What is the dimension of $f[x]$ over $f$

Let $f[x]$ be the ring of polynomials in one variable $x$ over the field $f$ with the relation $x^n =0$, for some fixed $n \in \mathbb{N}$. How can I find the dimension?
-1
votes
1answer
47 views

Suppose$V_1,\ldots,V_m$ are vector spaces. Prove these two vector spaces are isomorphic [closed]

The two vector spaces are: $L(V_1×\cdots ×V_m,W)$ and $L(V_1,W)\times\cdots\times L(V_m,W)$. Where $L(V,W)$ denotes the set of all linear maps from $V$ to $W$. Please help me with a rigorous proof, ...
1
vote
2answers
37 views

The function is not continuous

$$C([a,b])=\{ f: [a,b] \to \mathbb{R} \text{ continuous} \}$$ $C([a,b])$ is a linear space. For $f \in C([a,b])$ we define $\|f\|_{\infty}:= \sup_{x \in [a,b]} |f(x)|$ and easily it can be shown ...
0
votes
1answer
28 views

vector space constructed through a torsion module

Let $R$ be a principal ideal domain, $p \in R$ a prime element and $M$ a finitely generated $p$-torsion module of the form: $$ M = R/(p^{e_1}) \oplus \dots \oplus R/(p^{e_t}). $$ Let now be $_pM = ...
1
vote
1answer
25 views

a bilinear form is always the sum of two others

Let $K$ be a field with a characteristic, other than 2. Let $V$ be a finite dimensional vector space over $K$, and let $\gamma: V \times V \to K$ be a bilinear form. I now want to show that there ...
0
votes
0answers
41 views

What is an affline manifold? How do check if a space (?) is affline manifold. Is it related to vector space?

I have no idea about differential geometry or topology because I am engineer. I need the conditons to check the applicability of a theorem to particular case.From my search I could only find a wiki ...
1
vote
1answer
85 views

Detect when two edges make a “inner” angle or an “outer” angle

So, given three points, a direction of movement and the side of the movement, find out the "external" or "internal" angle value. In the left pic, I'm above the red line, moving from edge 1 to edge ...
0
votes
1answer
16 views

Additive function in $\mathbb{R}^n$ is continuous, and related subspaces compact

I want to show that the function: $A: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}^n, (x, y) \mapsto x + y$ is continuous. Also, why is it that if $K, L$ are compact subspaces of $\mathbb{R}^n$, ...
0
votes
1answer
21 views

Linear Span of R3

I am stuck with this question from my assignment in which its given that W1 = L{(1,1,0),(-1,0,2)} and W2 = L{(1,0,2),(-1,0,4)} and it being asked to show that W1 + W2 = R3. Following are my ...
2
votes
5answers
84 views

Do planes stop, or are they ever expanding?

I am trying to understand sub-spaces in linear algebra and one of the rules mentions if W is my subspace then if k is any scalar and u is any vector in W then ku is in W. I am unsure how this works ? ...
0
votes
1answer
43 views

Does an orthogonal decomposition of a vector space exist?

Let V be a complex vector space equipped with an hermitian form (not necessarily positive definite), W a finite dimensional subspace of V such that it has zero radical (intersection between W and its ...
2
votes
3answers
30 views

Base of the $\mathbb{R}$ vector space that contains all real functions: $f(x) \not= 0$ for finitely many x $\in\mathbb{R}$

I did already prove that this is a vector space. It is easily shown that addition and scalar multiplication with functions that hold the above property again yields a function with $f(x) \not= 0$ for ...
1
vote
2answers
22 views

linear independency in equation of linear span

we got the following vectors: $$v_1, v_2, w_1, w_3 \in V$$ $V$ is a vector space so that $\DeclareMathOperator{Sp}{Sp}\Sp\{v_1,v_2\} = \Sp\{w_1,w_2\}$ it's also defined that $\{v_1,w_2\}$ is linear ...
1
vote
1answer
44 views

what represents this strange matrix [closed]

But I just encountered a very strange type of matrix. What is the name of this type of matrix? How can it be solved? The final result should be a $4\times 4$ matrix, but I only see $5$ elements on ...
0
votes
1answer
26 views

$L\cap\left(M+\left(L\cap N\right)\right)=\left(L\cap M\right)+\left(L\cap N\right)$

I need to prove $$L\cap\left(M+\left(L\cap N\right)\right)=\left(L\cap M\right)+\left(L\cap N\right)$$ RHS is in LHS: it's easy to see that each of intersection of RHS is in LHS. How to make ...
0
votes
0answers
19 views

The difference between a basis of vectors vs functions

If my understanding is correct, I can see that for any set of linearly independent vectors $V = v_1, v_2, ..., v_n$ these establish a basis for $R^n$ by way of a set of coordinates $X = ...
0
votes
0answers
16 views

An electron in a TV tube is beamed horizontally at a speed of 4.3 * 10^6 meters per second

An electron in a TV tube is beamed horizontally at a speed of 4.3 * 0^6 meters per second toward the face of the tube 31 cm away. How far will the electron drop before it hits? (Assume ideal ...
1
vote
1answer
17 views

On the characterization of a subspace (proof of a lemma).

This is a very well know lemma I am trying to prove this. I have some doubts on proving that $W$ has the identity element of addition, i.e., that there exists an element $0 \in W$ s.t. $\forall w ...
3
votes
1answer
125 views

A name for the property $ \| x \star y \| = \| x \| \| y \| $.

Suppose that $ \star: V \times V \to V $ is some binary operation on a vector space $ V $. Should it hold, is there a name for the following property? $$ \forall x,y \in V: \quad \| x \star y \| = \| ...
9
votes
3answers
167 views

What do groups and rings “look like”?

Taking undergraduate physics courses, I had to deal with Euclidean vectors often. In classes like Calc III, the concept was also there. I'm not sure if this is why, but I've always had a more ...
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
29 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
33 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
31 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
29 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
26 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
43 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
23 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
42 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?
2
votes
1answer
28 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
28 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
54 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
36 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
32 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
748 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
19 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
18 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
147 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
91 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
37 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
26 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
28 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
92 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
37 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. ...