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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1answer
39 views

Vector spaces and linear dependence?

I'm struggling with this question: Find $\alpha$ such that the set of vectors is linearly dependent. $$\begin{bmatrix} 1\\ 2\\ 7 \end{bmatrix},\begin{bmatrix} 2\\ 11\\ -5 \end{bmatrix},\begin{...
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0answers
77 views

Can we say anything about the relationship between these functors?

I am working with a category $\mathcal{C}$ and two functors $F:\mathcal{C}\rightarrow \mathbb{R}$-$\mathbf{Vect}$ and $G:\mathcal{C}^{\operatorname{op}}\rightarrow \mathbb{R}$-$\mathbf{Vect}$ where $\...
6
votes
1answer
303 views

Calculating the intersection of two spaces of polynomials

This problem is driving me nuts. I feel like there should be an elementary argument, yet I have failed to find one. Consider the vector space $V_n=\mathbb Q[x]/{x^{2n+1}}=\mathbb Q\{1,x,x^2,\ldots, x^{...
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2answers
692 views

How to find a linearly independent vector?

Given two vectors $(1,2,8),(0,1,9)$ find a 3rd vector that is linearly independent from these two vectors. I sort of have an idea how to go about solving the problem but I'm not 100% sure. I'm know ...
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1answer
40 views

Vector basis answer/method check?

Question: Given a spanning set of a subspace $U\subset \mathbb{R^3}$ where $U=\{(-1,-5,0),(-4,-21,-4),(1,10,20),(2,12,8)\}$ find a basis of $U$. Solution I first setup a matrix where each row was ...
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1answer
28 views

equation by parts?

I was doing vectors and encountered an exercise in which it says ''write down the equation by parts of the plane'' what does it mean ''equation by parts of the plane''?
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0answers
246 views

Find the zero vector of a vector space of $2\times 2$ matrices called $M$

For my last assignment of linear algebra, I have an exercise where I have to find the zero vector of a vector space, but I am having some problems understanding what I really have to answer. This is ...
5
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0answers
87 views

Mapping vector spaces over two different fields?

I was having linear algebra class and we have been discussing about a possible group homomorphism that might allow mapping between two vector spaces over two different fields This is also an ...
2
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2answers
63 views

Seeing the plane as a four (or more) dimensional vector space on $\mathbb Q$

As I was trying to answer a question about the enumeration of circuits one can build with a set of miniature train track elements, I realized that all plane positions that could be reached had ...
0
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4answers
73 views

Let $V = \mathbb{R}^2$, show that $V$ is a vector space

I am new to the concepts of vector spaces (as far as I remember), and I have some difficulties in understanding how can I show, in practice, if a set is a vector space or not. I have an exercise in ...
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2answers
25 views

Show that there are no values of a such that span{$u_1,u_2,u_3$} is a line in $\mathbb{R}^3$ that passes through the origin.

Let $u_1=(1,-1,a)$, $u_2=(a,0,1)$, $u_3=(1,1,a)$. (a) Show that there are no values of a such that span{$u_1,u_2,u_3$} is a line in $\mathbb{R}^3$ that passes through the origin. I figured out that $...
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0answers
31 views

$R$ be a commutative ring which is a vector space over some field $F$ , is the map $f(x)=rx , \forall x \in R$ $F$- linear for every $r \in R$?

Let $R$ be a commutative ring which is a vector space over some field $F$ ; for $r \in R$ consider the function $f:R \to R$ defined as $f(x)=rx , \forall x \in R$ , is this function a vector space ...
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2answers
37 views

Two vectors moving towards the same point - ensuring they both hit that point at the same time

I'm working on an algorithm which involves two vectors in 3D space. They're both moving towards a single point within their respective directions - I need to make sure that they both hit the same ...
1
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0answers
32 views

In the space of polynomials of degree 2 or less, given the derivative linear transformation D and $T:=1+D+D^2$, $S:=1-D$, show that $S=T^{-1}$

Let $ P_2[X] $ be the space of polynomials of degree equal or less than 2 over the field R. Let: $$ D: P_2[X] \rightarrow P_2[X] $$ Be the derivative linear transformation, defined as follows: $$ (D(P)...
0
votes
1answer
108 views

What this notation R^3 ∖ (0, 0, 0) means?

I was reading a "Projective Space" article on Wikipedia, when I came across this line "equivalent definition is the set of equivalence classes of $\mathbb R^3 \setminus (0, 0, 0),$ i.e. 3-space ...
1
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2answers
91 views

About a norm : $p(uv)=p(u)p(v)$ all the time? [closed]

Say, $p$ is a norm on a vector space. Then can we say that $$p(uv)=p(u)p(v)$$ all the time? Thanks.
1
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2answers
74 views

Determine whether each of $V_1$, $V_2$, $V_3$ is a subspace of $\mathbb{R}^4$

Let $V_1=\{(x,y,z,w)\space|\space x-y+z=2w\}$ $V_2=\{(x,y,z,w)\space|\space xyz=2w\}$ $V_3=\{(x,y,z,w)\space|\space x-y+z=2^w\}$ (a) Determine whether each of $V_1$, $V_2$, $V_3$ is a subspace of $...
1
vote
1answer
34 views

On the image of some linear maps

Let $V$ be an $n$-dimensional vector space with two endomorphisms $f,g:V\to V$. I have a linear map \begin{align} \psi_{f,g}:V\times V&\to f(V)\times g(V)\subset V\times V\notag\\ (v,w)&\...
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0answers
80 views

Basis of the vector space generated by a convex cone

Let $C$ be a pointed convex cone in a vector space, meaning that: $C + C \subset C$, $\mathbb{R}_+ \cdot C \subset C$, $C \cap (-C) = \{ 0 \}$. If $V$ denotes the vector space generated by $C$,...
1
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1answer
48 views

Probability that two subspaces are complementary

$p$ - a prime number $Gr_{n,k}(F_p)$ - set of all k-dimensional subspaces of the vector space $F_p^n$ $F_p$ - finite field with p elements For $1\leq k\leq n-1$ fixed, the subspaces $V$ and $W$ are ...
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0answers
19 views

linear span of set of independent vectors

for basis of vector , we need to prove that a set of vectors is contained in that space and should be linearly independent .But how to show that linear span of that set of linearly independent vectors ...
0
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2answers
33 views

Show that $|x-y|<0.5|y| \implies |x|>0.5|y|$

How to show that $|x-y|<0.5|y| \implies |x|>0.5|y|$, where $|...|$ is a norm and $x,y$ are vectors of a vector space over field $R$?
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2answers
59 views

Finding dimension of vector spaces

Let V be a vector space and W a subspace of V . Let q : V → V/W be defined by q(v) = v + W for v ∈ V. a) Prove that q : V → V/W is a linear transformation which is onto and show that N(q) = W. b) ...
2
votes
1answer
30 views

Why is the rank of a matrix capped in this way?

Take a $3\times5$ matrix $A$ then we have that $rank(A)\leq3$. More specifically we have that when we think about the rows and columns of $A$ as vectors then any collection of more than three $3$‐...
2
votes
3answers
56 views

Question about the relationship between subspaces and dimension in vector space.

Hello I'm wanting to answer this question but I cannot figure out really what to do. Suppose $\overrightarrow{v_1},\overrightarrow{v_2},\overrightarrow{v_3},\overrightarrow{v_4}$ are vectors in a ...
0
votes
1answer
31 views

What does this set of vectors describe geometrically speaking?

What does this set of vectors describe geometrically speaking? $$\{(x,y,z):2x=-6y=4z\}$$
3
votes
0answers
28 views

Is this the correct solution involving vector subspaces and basis?

I need to find the basis and hence dimension of a subspace of $\mathbb{R^3}$. 1) $$U=\{(x,y,z):x=2y\}$$ Solution: We have $x=2y \iff y=\frac{x}{2}$ therefore we can write all elements in $U$ as the ...
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2answers
49 views

About vector spaces

Show that if a vector space contains two elements, then it contains infinitely many. I have a question to proove but my brain couldnt work this out can you please show how to prove this argument?
1
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1answer
153 views

How can I move a point along a line in 3D space to reach a target dot product with a fixed reference point?

Suppose a point in 3D space, Q. For any other point x in that space, Let Q(x) be the unit vector pointing from x towards Q. I also have a line L in 3D space, and a point on this line P. L = {P + k*...
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0answers
36 views

Sequence subspace

Wondering about this exercise: 1)The sequence subset $(u_n)_{n\in\mathbb{N}}$ such as $\forall n \ \in \mathbb{N^*}, u_{n+2}=u_n+u_{n-1}$ is it a E-subspace ? $0=(0)_n \ \in \ F $ because 0=0+0 but ...
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1answer
18 views

Elementary question subspace

Wondering about the difference here: Let $E=(z \ \in C; Re(z)=0)$. Is it a subspace of $\mathbb{R}$-vecspace$\ \mathbb{C}$ and a subspace of $\mathbb{C}$-vecspace $\mathbb{C}$ ? I easily show it is ...
2
votes
4answers
29 views

Polynomial subspace

Wondering abut this set : $E=(p(X) \ \in \mathbb{R}[X]; Xp(X)+p'(X)=0)$, is it a subspace of $\mathbb{R}[X] $? I definitiley think it is because it only includes the zero polynomial but how could we ...
1
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1answer
46 views

Show that the step function is linearly independent

Consider the set V consisting of all functions $f : \mathbb R \to \mathbb R$, considered as a vector space over $\mathbb R$ with the usual definitions of addition and scalar multiplication. Consider ...
1
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1answer
35 views

Basic subspaces

I'm beginning studying vectorial spaces and I'm wondering about theses spaces: $\text{Among these subsets of} \ \mathbb{R^3}, \ \text{which one are subspaces ?}$ 1) $E{_1}= ((a+b,a-b,2a); \ a,b \in ...
3
votes
0answers
124 views

Matrix which is not similar to it's transposed

Let $V$ be vector space over a field $\mathbb{k}$. I can prove that any matrix is similar to its matrix transpose if $\mathbb{k}$ is an infinite field, but is this still true when $\Bbb k$ is finite? ...
0
votes
1answer
29 views

How to find a basis with 2 constraints

If V is a subspace with $(x_1,x_2,x_3,x_4)\in R^4$ such that $x_1 -2x_2+x_3=0, 2x_1-3x_2+x_3 = 0$ How would I find a basis for this? I cant seem to find a way other than inspection because normally I ...
0
votes
1answer
136 views

Finding a basis for a vector space with equations

Let $V = \{ (x_1, x_2, x_3,x_4)\in R^4: x_1-2x_2+x_4 = 0, 2x_1-3x_2+x_3 = 0 \}$ I am trying to find a basis for V. Subtracting the constraints from each other yields $x_4= x_2/2+x_3/2$ this means ...
2
votes
1answer
70 views

What does $E^{\mathbb{C}}$ mean?

I was reading a book (Symplectic Geometry and Quantum Mechanics) and find it hard to understand this following example: Definition: a "complex structure" on a vector space $E$ is any linear ...
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votes
0answers
33 views

Finding the basis of a vector space $\Bbb{W}$ [duplicate]

Given that $\Bbb{W}=\{(a,b)|a,b\in\Bbb{R}\}$ with addition defined by $(a,b)\oplus(c,d)=(a+c+1, b+d)$ and scalar multiplication defined by $k\odot(a,b)=(ka-k+1, kb)$ is a vector space, find a basis ...
1
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1answer
125 views

Finding out if vectors are Parallel or Orthogonal in Parametric Form.

I have two Parametric Form Vectors. Is it possible in that form to work out if the vectors are Parallel, Orthogonal or neither. Or do I have to have it in standard vector form $ (a_1, a_2, a_3)$ and ...
0
votes
2answers
28 views

If $\mathcal{B}$ is a basis for a $\Bbb{V}$ and $\Bbb{U} \subseteq \Bbb{V}$ then $\exists\mathcal{S} \subseteq \mathcal{B}$ basis for $\Bbb{U}$

If $\Bbb{U}$ is a subspace of a finite dimensional vector space $\Bbb{V}$ and $\mathcal{B}=\{\vec{v}_1,..,\vec{v}_n\}$ is a basis for $\Bbb{V}$, then some subset of $\mathcal{B}$ is a basis for $\Bbb{...
0
votes
3answers
83 views

Finding a basis for all $2\times2$ matrices A such that…

Finding a basis for all $2\times2$ matrices A such that $$\left[ \begin{matrix} 1 \hspace{5pt} 2 \\ 0 \hspace{5pt} 3 \end{matrix} \right]A=\left[ \begin{matrix} 0 \hspace{5pt} 0\\ 0 \hspace{5pt} 0 \...
0
votes
2answers
335 views

Given a Transformation Matrix $T$, find $T$ relative to a new basis $\beta$

$T(a_1,a_2,a_3) = (3a_1+a_2,a_1+a_3,a_1-a_3)$. $(a_1,a_2,a_3)^T$ is written with regards to the standard basis. We can figure out $T$ in matrix form by calculating $T(a_1),T(a_2), T(a_3)$. That's ...
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1answer
679 views

Finding the basis of all vectors perpendicular to one vector

Let v = $\begin{bmatrix} 1 \\ 2 \\ 3 \\ 4 \\ \end{bmatrix}$ in $\mathbb R^4$. How I can find a basis of the subspace of $\mathbb R^4$ such that the subspace ...
2
votes
3answers
54 views

Rewriting vector as sum of components in the subspaces $V$ and $V^\perp$

I came across this problem: how can I write vector $u = (4,3,5)$ as $u=v+w$ where $v \in V = \operatorname{span} \left \{ (1,1,0),(0,1,1)\right \} $ and $w \in V^\perp$. Thanks for advice.
19
votes
4answers
669 views

Why the whole exterior algebra?

So, I've been reading up on multilinear algebra a bit. In particular, I've been reading up on the construction of of the exterior algebra of a finite dimensional vector space $X$, say over $\mathbb{R}$...
0
votes
2answers
24 views

make a vector with variable a linear combination

I have this 3 vectors: $\overrightarrow{u} = \{1,3,a\}$ $\overrightarrow{v} = \{1,-1,0\}$ $\overrightarrow{w} = \{2,1,1\}$ I need to find the 'a' in vector $\overrightarrow{u}$ such that $\...
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votes
4answers
29 views

calculate '$a$' so that the vectors are a linear combination

How to calculate the value of 'a' so that $$ \overrightarrow{u} = \{1,3,a\} $$ is a linear combination of $$\overrightarrow{v} = \{1,-1,0\} $$ and $$ \overrightarrow{w} = \{2,1,1\} $$
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votes
1answer
26 views

How to find the $k$ value in this question?

Calculate $k$ such that $u=(1,1,1)$, $v=(k,0,1)$ and $w=(2,-1,-2)$ are linearly dependent, but are pairwise linearly independent.
3
votes
1answer
49 views

is $V$ Zariski dense in $K\otimes V$?

Suppose $V$ is a vector space over infinite field $k$ and $K$ an extension of $k$. If we consider $(V)_K= K \otimes V$ as a $K$-vector space then in what sense $V$ is Zariski dense in $(V)_K$?