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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2answers
24 views

rotation matrix and vector - understand step calculation

I have an extremely equation, but I just don't understand which step they made to get to the last line. ${\bf W}$ and ${\bf V}$ are all 3d vectors. A is a rotation matrix. How did they get that ...
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1answer
23 views

In $\mathscr{V}$, let $X \subset \mathscr{V}$ be a set of $n$ vectors. $Y \subset X$ contains vectors all scalar multiples, $X$ linearly dependent.

I would just like to verify that my proofs are sound and receive any suggestions on rewording. (If relevant, I am self-studying and haven't done a serious proof in about a year.) $\mathscr{V}$ is a ...
1
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3answers
41 views

Understand the definition of a vector subspace

I'm pretty new to Linear Algebra and I have started on Vector Spaces. I understand that a Vector space V over the set of real numbers is a set equipped with two operations, namely vector addition and ...
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3answers
10 views

Give two matrices whose column spaces contain the column space of the given matrix.

Let $$B = \begin{pmatrix} 1 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{pmatrix}\text{.}$$ Give two matrices whose column spaces contain $C(B)$, the column ...
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3answers
35 views

Show that a linear transformation $T$ is one-to-one

Problem: Consider the transformation $T : P_1 -> \Bbb R^2$, where $T(p(x)) = (p(0), p(1))$ for every polynomial $p(x) $ in $P_1$. Where $P_1$ is the vector space of all polynomials with degree less ...
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2answers
25 views

Finding a Basis for this subspace

Set $V=\mathbb{R}^{2x3}$ and let $U$ be a subspace of $V$ defined by: \begin{equation*} U=\{B=(b_{ij})\in V\mid b_{11} + b_{12} + b_{13} = -4(b_{21} + b_{22} + b_{23})\}. \end{equation*} I would ...
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1answer
20 views

What's the detailed derivation from equation 126 to 127 (See Figure)? (problem of Lagrange method with vector transpose)?

I am confused of vector transpose, I do not know how to get equation 127. Thanks.
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2answers
22 views

Norm is sup of inner products (proof).

Let $V$ be a vector space with an inner product $\langle.,. \rangle$ and associated norm $|| . ||$ Then: Could I have a proof of this fact?
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1answer
23 views

Covariant metric tensor of a subspace

Suppose $f_1,f_2$ and $f_3$ are vectors in a vector space $V$ with a dot product. Me assume that the vectors are linearly independent. What does it mean to find the covariant metric tensor of ...
0
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1answer
33 views

Find a matrix whose column space contains the column space of the given matrix.

Let $$A = \begin{pmatrix} 1 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 1 \end{pmatrix}\text{.}$$ $C(A)$ denotes the column ...
0
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1answer
35 views

Do the spaces spanned by the columns of a matrix and by the columns of a set of matrices coincide?

As in Do the spaces spanned by the columns of the given matrices coincide?, let $$A = \begin{pmatrix} 1 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ ...
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1answer
23 views

Uncountable “relatively independent” subset of finite dimensional vector spaces over an uncountable field

Let $V$ be a $n$ dimensional vector space over an uncountable field ; then does there always exist an uncountable subset $S$ of $V$ such that any $n$ vectors of $S$ are linearly independent ? ( I can ...
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2answers
36 views

Proof with subspaces [on hold]

Prove: If $V$ and $W$ are three-dimensional subspaces of $\Bbb R^5$, then $V$ and $W$ must have a non-zero vector in common. (Hint: start with bases for the two sub-spaces, making six vectors in all) ...
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4answers
36 views

For every integer vector $\overrightarrow{a}$,there is a integer vector $\overrightarrow{b}$ such that $\overrightarrow{a}\bot\overrightarrow{b}$

In $R^3$,show that for every integer vector $\overrightarrow{a}$,there is a integer vector $\overrightarrow{b}$ such that $\overrightarrow{a}\bot\overrightarrow{b}$ Generally,in $R^n$,for every ...
2
votes
2answers
22 views

Faithful module is infinitely dimensional as a vector space.

Let $M$ be a $\mathbb{C}[x]$-module. Since $\mathbb{C} \subset \mathbb{C}[x]$, we may consider $M$ as a $\mathbb{C}$-vector space. If $M$ is faithful, must $M$ be infinite dimensional as a ...
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1answer
21 views

Check for basis of a matrix

Given the matrices in $M_{3,3}$. ...
2
votes
2answers
30 views

Do the spaces spanned by the columns of the given matrices coincide?

Reviewing linear algebra here. Let $$A = \begin{pmatrix} 1 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 1 \end{pmatrix} \qquad ...
1
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1answer
23 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?
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2answers
41 views

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

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, ...
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0answers
30 views

What are Products and Quotients of vector space used for?

I'm self-studying the book Linead Algebra Done Right, and I'm now confused by these two space. What's the motivation for defining them? What are they used for? What's the insight of them? Please help ...
1
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2answers
30 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 ...
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1answer
14 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
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1answer
20 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 ...
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0answers
26 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
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1answer
25 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 ...
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1answer
15 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
19 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
81 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
36 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
27 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
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2answers
16 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
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1answer
33 views

space engineering - strange matrix - attitude [on hold]

I am in charge with developing software to determine the satellites attitude. But I just encountered a very strange type of matrix $K$ (from the $q$-method attitude determination algorithm) What is ...
0
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1answer
21 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 ...
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0answers
17 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 = ...
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0answers
14 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
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1answer
15 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
105 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
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3answers
142 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
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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
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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 ∈ ...
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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
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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 ...
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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
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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
23 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
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2answers
18 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?
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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 ...