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

0
votes
1answer
26 views

Direct Sum of Three Subspaces

Suppose $U = \{(x, y, x+y, x -y, 2x) \in \Bbb F^5 : x, y \in \Bbb F\}$. Find three subspaces $W_1, W_2, W_3$ of $\Bbb F^5$, none of which equal $\{0\}$ such that $\Bbb F^5 = U \oplus W_1 \oplus W_2 ...
0
votes
2answers
34 views

vectors in a space (very simple question)

My question is too stupid to be googled, so I'll ask it here (because I didn't get any answers from google). Context: I have a three dimensional space and the units for x,y,z are given in meters. On ...
0
votes
0answers
18 views

Minimum in complex inner product vector space

I'm stuck at this problem, can someone give me a hint? Let $x_i$ and $y_i$ ($i=\overline{1,n}$) be vectors in an infinite dimensional vector space $V$ with inner product $(,)$ satisfy: ...
1
vote
1answer
9 views

Finding the conditions of (x,y,z,t) for them to belong to the span of a set of vectors

So I got this math exercise, and I don't know how to go about it: In $\mathbb{R}^4$, $S$ is the subspace spanned by the following set of vectors: $(1, 1, 1, 0) , (1, 2, 1, 1) , (2, 0, 1, 1) , (3, 0, ...
1
vote
0answers
29 views

Coordinate matrices in standard basis

If $A=1+2x+4x^3$ and $B=2+3x^2+x^3$ are vectors in Polynomial space, find out the coordinate matrices for $A$ and $B$ in standard basis and hence find out the angle between vectors A and B.
1
vote
0answers
22 views

Is there a computationally efficient way to find the part of a vector, which is of certain order in independent variable x?

Let $\vec{a}$ be an element of a vector space over the space of monomials, i.e. $$ \vec{a}\left(x\right)=\sum_{j=1}^{N}a_jx^{k_{j}}\vec{e_{j}} $$ Remark: For simplicity, here we operate with only ...
0
votes
3answers
24 views

How to prove a levi-civita symbol and kronecker delta relationship [duplicate]

When I do the calculations of that I get 3 times the answer, I mean this is easy, but I´m just wrong, Could someone show me the way?
0
votes
0answers
17 views

$\dim \mathcal{S}_k(\Gamma_0(N))$

I'm looking for a formula which gives the dimension of $\mathcal{S}_k(\Gamma_0(N))$ the space of cusp forms of weight $k$ and level $N$. I found the following statement for $k\geq 4$ $$\dim ...
0
votes
0answers
18 views

Finding the exponential relation between two 4x4 transition matrices

Im alright with matrices, but this question has dumb-struck me. Suppose I have two known and given $4\times4$ transition matrices, representing transitions in three dimensions with the fourth ...
1
vote
2answers
17 views

Prove that addition of a constant on vector spaces is bijective

What would be a nice way to deduce from the vector space axioms that $f : V_1 \longrightarrow V_2, \, x\mapsto x+v$ with constant $v$ is bijective?
0
votes
0answers
12 views

Projection of vector onto intersection of two planes

Given that $U$ is a vector space in $\mathbb{R}^{n}$ there exists a unique vector $\mathbf{u_{0}}$ such that $$\left \| \mathbf{v}-\mathbf{u_{0}} \right \|\leq \left \| \mathbf{v}-\mathbf{u} \right ...
0
votes
0answers
13 views

Properties of cross product ${\rm i}(a\times a^*)$

Given a complex 3-vector $a\in\mathbb{C}^3$, let $b$ be the following vector $$b={\rm i}(a\times a^*)$$ where $a^*$ is the element-wise complex conjugate of $a$. As can be easily shown by ...
1
vote
1answer
16 views

Vector space and its Projecctivized Space

Why is the co-dimension one subspaces are the points of $\mathbb P(V^{\vee})$. $V^{\vee}$ is the dual space of V and and $\mathbb P(V)$ is the projectivized space of V. $\mathbb P(V)= ...
0
votes
0answers
15 views

Do the Generalized Gell-Mann Matrices form a complete set?

Please bear with me, I'm studying Lie algebras as they are related to quantum mechanics, and most of my group theory knowledge is self-taught. I'm not sure how to prove this seemingly basic result. ...
2
votes
2answers
40 views

Dimension of vector space in extreme cases.

Let V be a vector space of dimension 29 over a field $\mathcal{F}$. Suppose that U and W are subspaces of V with dim(U) = 24 and dim(W) = 15 1) What are the possible values of dim(U+W)? My ...
4
votes
2answers
82 views

Localization does not commute canonically with infinite direct products

Let $S=\mathbb{Z}-\{0\}$, and the fraction ring \begin{equation} S^{-1}\prod_{1}^{\infty}\mathbb{Z}_{i}=\{\frac{(a_{1},a_{2},...,a_{n},...)}{b}:b,a_{i}\in\mathbb{Z},b\neq 0\}.\end{equation} Show ...
2
votes
0answers
22 views

How to put a structure of Fréchet space on $\Gamma(E)$?

Let $\pi: E \to M$ a smooth vector bundle over M. If $(M,g^{M})$ and $(E,g^{E})$ are complete manifolds. Consider $\nabla$ a conection on $E$. We can define these semi-norms ...
1
vote
0answers
25 views

Getting coordinate vector in linear algebra

I know how to get the coordinate vector of single matrices by just joining them and doing a gauss jordan. But these are a 2x2, I don't know how to go about this, apparently no elimination can take ...
2
votes
0answers
36 views

How many (unordered) bases does $\Bbb F_q^n$ have as a vector space over $\Bbb F_q$?

This is a practice problem from a Tier 1 exam, and I want to check that my reasoning is correct. We shall first consider how many different ordered bases $\Bbb F_q^n$ has. Recall that $|GL_n(\Bbb ...
0
votes
2answers
29 views

vector question assistance

let there be 2 lines: $(2,-3,1) + s(3,-2,1)$ and $(2,-1,-3) +t(3,-2,1)$ which are parallel to each other. find the formula of the plane determined by them. my try: a vector perpendicular to ...
0
votes
1answer
19 views

How to show a vector space is not closed under addition with elements not in the vector space.

Wasn't entirely sure how to word the title. What I'm trying to show is: Given $\vec{v}\in V$ and $\vec{w}\not\in V$, then $\vec{v}+\vec{w}\not\in V$ How would this statement be proven?
0
votes
0answers
9 views

Difference between containing point and pass through point?

I do not understand this, What is the difference between the equation of the plane containing the points and the equation of the plane through the point? Is it the same thing or are they different?
1
vote
0answers
21 views

In Exercises1–6, find a basis for the solution space of the ho mogeneous linear system, and find the dimension of that space.

for number 1, I tried reducing the matrix to solve for x1, x2, x3, but i got a row of 0's, so I'm not sure what to do after that... and what exactly if i have a solution? i saw an example of having ...
3
votes
2answers
39 views

Surjective linear transformations: is checking for linear independency enough?

Is the linear transformation $T: \mathbb{R}^3\to\mathbb{R}^3$ defined as $$\begin{bmatrix} 0 & 1 & 0 \\ 0 & 1 & 0 \\ 1 & 1 & -1 \end{bmatrix}\begin{bmatrix} x \\ y ...
0
votes
4answers
45 views

algebraic representation of a line in 3d

Is an algebraic representation of a line in 3d possible, or there can be only a parametric one?
0
votes
1answer
36 views

Show that $\lambda_1 = \min \{ Q(u) \mid \|u\| = 1 \}$ and $\lambda_m = \max \{ Q(u) \mid \|u\| = 1 \}$

Let $V$ a vector space over $K$ and $Q(u) = \langle u, Tu \rangle$ a quadratic form. $T$ is a symmetric operator. The eigenvalues of $T$ are sorted by size $\lambda_1 < \dots < \lambda_m$. How ...
0
votes
2answers
42 views

Vector problem (line on a plane)

For which values of $a$ and $b$ is this line $$\vec{r}= \begin{pmatrix} 3\\2\\a \end{pmatrix}+\lambda \begin{pmatrix} 2\\b\\1\end{pmatrix} $$ inside of the plane $x-y+2z=11$ ? I'm totally missing ...
1
vote
1answer
29 views

Calculating an orthonormal base given another base.

Let $$W = \operatorname{span}(\{(1,1,1),(0,0,1)\})$$ Find an orthonormal base $B$ of $W$. So. An orthonormal set is a base whose elements are orthogonal with each other and their length is ...
1
vote
1answer
44 views

What is the difference between $\mathcal{X}\subseteq\mathbb{R}^n$ and $\mathcal{X}\subset\mathbb{R}^n$

Let $\mathcal{X}$ be a non-empty set. For instance, let it be a set of vectors of the form $\mathbf{x}\in\mathbb{R}^n$, i.e., ...
0
votes
0answers
17 views

computing equation with vectors

I've got this equation to compute. In fact i'd like to be able to compute every weight $w(k+1)$ knowing its past value $w(k)$. the equation is : $$\bf w\rm_{ij} (k+1) = \bf w\rm_{ij} (k) - ...
2
votes
1answer
21 views

Determining subspaces of $P_3$

Theorem: If $W$ is a set of one or more vectors in a vector space $V$, then $W$ is a subspace of $V$ if and only if the following conditions are satisfied. If $\mathbf{u}$ and $\mathbf{v}$ are ...
4
votes
1answer
35 views

Quotient $M/M^2$ is finite dimensional over $R/M$ in local Noetherian ring?

I have that $R$ is a Noetherian local ring with maximal ideal $M$, and I want to show that $M/M^2$ is a finite dimensional vector space over the field $R/M$. I think I've proved this (though I ...
0
votes
2answers
65 views

Is the determinant of a matrix some kind of “integral” of the linear mapping?

A $n \times n$ matrix corresponds to a linear mapping between two $n$-dim vector spaces. The determinant of a matrix gives a scalar, just as the integral of an integrable function gives a scalar. ...
1
vote
2answers
34 views

If $V$ is a vector space over $\mathbb{Q}$ and $\alpha \in \mathrm{End}(V)$ is a projection, show that $\mathrm{spec}(\alpha)\subseteq {0,1}$

If $V$ is a vector space over $\mathbb{Q}$ and $\alpha \in \operatorname{End}(V)$ is a projection, show that $\operatorname{spec}(\alpha)\subseteq \{0,1\}$ The following is what I have so far, I am ...
1
vote
2answers
55 views

Write Matrix $A$ to $A = \sum_{i=1}^{3} \lambda_i P_i$

Let $A$ be a Matrix: $$A = \begin{pmatrix} 1 & 0 & 3i \\ 0 & -3 & 0 \\ -3i & 0 & 1 \end{pmatrix}$$ Now I want to write $A$ as $$A = \sum_{i=1}^{3} \lambda_i P_i$$ I ...
1
vote
2answers
52 views

What is physical interpretation of dot product?

Consider two vectors $V_1$ and $V_2$ in $\mathbb{R}^3$. When we take their dot product we get a real number. How is that number related to the vectors? Is there any way we can visualize it?
2
votes
2answers
51 views

How to bring $5x_1^2 - 26x_1x_2 + 5x_2^2 + 10x_1 - 26x_2 = 31$ to the form $\langle x',Ax' \rangle = 1$

How can I bring $$5x_1^2 - 26x_1x_2 + 5x_2^2 + 10x_1 - 26x_2 = 31$$ to the form $$\langle x',Ax' \rangle = 1$$ where $x' = \alpha x + \beta$ where $\alpha \in \mathbb{R}^+$ and $\beta \in ...
0
votes
2answers
36 views

$V,W$ are linear spaces.$ U_1,U_2$ are subspaces of $V$.

$V,W$ are linear spaces.$ U_1,U_2$ are sub-spaces of $V$. Also given, $\dim U_1 = \dim U_2 = 2n$ and there is a linear transformation $T: V \rightarrow W$ also $\dim (U_1 + U_2) = 3n$ prove that: ...
2
votes
0answers
29 views

$U,W$ are sub-spaces of $R^4$

Given $U,W$ are sub-spaces of $R^4$.Also ,$dim(U \cap W^\bot) \ge 2$ ,and $(1,1,0,0),(0,0,1,1) \in U$ and $(1,0,0,0),(0,0,0,1) \in W$ Find bases for $W,W^\bot,U$. Well, because $W$ is a subspace of ...
2
votes
1answer
44 views

Vector calculation question

the points a b c d are concordantly ( 1,2,-3) , (-1,2,1) , ( 0,1,-2) , ( 2,-1,1) find formula of the plane going thorugh d and which is pararlel to plane abc calculate the volume of pyramid abcd. ...
0
votes
0answers
20 views

Vector pyramid question

Suppose we have a pyramid with two vectors known, as well as the angle between them...if we mutpliy their size by each other mutiply by the sine of the said angle and then by sixth, will we get the ...
1
vote
1answer
40 views

Finite-dimensional subspace normed vector space is closed

I know that probably this question has already been answered, but I'd like to present my attempt of solution. Let $(E,\|\cdot\|)$ be a normed vector space and let $F\subseteq E$ be a ...
0
votes
0answers
21 views

vector subspace of all real polynomials which are divisible with $x^2 + 1$

Show that the set of all real polynomials which are divisible with $x^2 + 1$ is a vector subspace of space of all real polynomials to 4th degree. Also find base and dimension of this subspace. I ...
0
votes
2answers
37 views

Calculate the dimension of $U = \{(x_1,x_2,x_3,x_4,x_5) : x_1+x_3+x_5=x_2+x_4=0\}$

In the vector space $V \subset \Bbb R^5$, considering the vectors $v_1,v_2,v_3$ $v_1 = (0,1,1,0,0)$ $v_2 = (1,1,0,0,1)$ $v_3 = (1,0,1,0,1)$ We have $V = \mathrm{span}(v_1,v_2,v_3)$ ...
2
votes
2answers
58 views

Prove $\dim W \ge 2$

Let $U_1, U_2, W$ subspaces of a finite dimensional vector space, such that: $U_1 \cap U_2 = \{0\}$ $U_1 \cap W \ne \{0\}$ $U_2 \cap W \ne \{0\}$ Show that $\dim W \ge 2$. ...
1
vote
1answer
38 views

Subspaces of the set of real valued functions over an interval.

Show that the integral of all continuous real-valued functions on the interval [0,1] equal to b $\in$ R is a subspace of $R^{[0, 1]}$ if and only if b=0. So I am assuming that because both the ...
4
votes
1answer
30 views

$rk(A)=n$ implies $rk(AB)=rk(B)$

Let $A \in Mat_{m\times n}(\mathbb{R})$ and $B \in Mat_{n\times p}(\mathbb{R})$. Assume $rk(A)=n$. Prove that $rk(AB)=rk(B)$. Lets start by proving $rk(B) \ge rk(AB)$. Indeed, since the ...
2
votes
2answers
52 views

Show $L_1 \subseteq L_2$ or $L_2 \subseteq L_1$

Let $L_1,L_2$, two subspaces of a finite dimensional vector space. Prove that if $\dim(L_1 + L_2) = 1 + \dim (L_1 \cap L_2)$ then $L_1\subseteq L_2$ or $L_2 \subseteq L_1$. Well, I've read a ...
2
votes
1answer
25 views

Let $L,M$, two subspaces of $V$. Prove $L\cup M \ne V$ [duplicate]

Let $L,M$, two subspaces of $V$ where $L,M\ne V$. Prove $L\cup M \ne V$ My Try: Obliviously, $L\cup M \subseteq V$. It's left to show $L\cup M \subsetneq V$. $L,M$ have bases as subspaces. ...
0
votes
1answer
33 views

Using coordinates in B to find coordinates in B'

Part 1: $B=\{v_1,v_2,v_3\}$ is a basis of the real vector space $V$. ${B'}=\{v_1+v_2,av_1+v_3, bv_1-v_3\},\quad a,b\in\mathbb R$ What conditions should $a$ and $b$ satisfy for $B'$ to be a ...