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
1k views

Cartesian vector to Cylindrical components

I recently started EE 1, and was confused by the jump from cylindrical coordinates to cylindrical vectors In the past, I've just converted between cartesian and cylindrical via: p = sqrt(X^2 + Y^2) ...
1
vote
1answer
91 views

Function Vector Spaces: Set to Field

I'm having trouble answering the last problem Linear Algebra set. Not looking for a solution, of course, but some pointers would be incredibly helpful. Given a vector space $F^S$ of all functions ...
0
votes
2answers
80 views

Vectors: difference between $ab$ and $a^Tb$

I have been given a vector problem, np as I am good with vectors. But I was educated in Denmark, and I'm currently in America. The assignment is Find $a^T\cdot b$. Now I have never seen this ...
2
votes
1answer
2k views

How to prove the sum of 2 linearly independent vectors is also linearly independent?

Suppose $a,b$ and $c$ are linearly independent vectors in a vector space $V$. How can I prove that $a+b$ or $b+c$ are also linearly independent?
0
votes
1answer
2k views

Parametric equation of line parallel to a plane

The parametric equation of the line is $$x=2t+1, y=3t-1,z=t+2$$ The plane it is parallel to is $$x-by+2bz = 6 $$ My approach so far I know that i need to dot the equation of the normal with the ...
0
votes
2answers
104 views

Vector cross product and dot product.

Given $a = \langle 1,-1,2\rangle$ and $b = \langle 2,1,0\rangle$ . Find $t$ such that the vector $c = \langle 5,t-1,2\rangle$ is perpendicular to $a \times b$.
1
vote
1answer
68 views

Proof that $\operatorname{span}(S_1\cap S_2) \subseteq \operatorname{span}(S_1) \cap \operatorname{span}(S_2)$

An exercise in my book says : Prove that $\operatorname{span}(S_1\cap S_2) \subseteq \operatorname{span}(S_1) \cap \operatorname{span}(S_2)$. Give an example in which $\operatorname{span}(S_1 ...
-1
votes
3answers
169 views

Possible Cardinality of a Field

The following question struck me as pretty interesting: Let $\Bbb F$ be a field of characteristic $p$ (a prime, of course). I'm then asked to show that $|\mathbb{F}| = p^n$ for some $n\geq 1$. ...
1
vote
2answers
64 views

Verifying a bijection

Let $V$ be vector space over $\mathbb{F}$, and $W\subseteq V$ a subspace. Let $p:V\rightarrow V/W$ be the canonical projection. Let $X$ be the set of all subspaces containing $W$ and $Y$ be the sets ...
2
votes
2answers
80 views

Proving we have a basis for $F[x]$

So $F$ is an arbitrary field, and $F[x]$ denotes the set of of formal polynomials with coefficients in $F$. And $A=\{f_i \mid i\geq 1\}$. I need to show two things, If $A$ is such that $deg (f_i) ...
0
votes
4answers
103 views

Proof that $V = W_1 \oplus W_2$

One of many assignments is: Let $W_1$ and $W_2$ be subspaces of a vector space $V$. Prove that $V$ is the direct sum of $W_1$ and $W_2$ if and oly if each vector in $V$ can be uniquely written as ...
1
vote
2answers
739 views

Where is the difference between the union and sum of sets?

My book writes: Definition. A vector space $V$ is called the direct sum of $W_1$ and $W_2$ if $W_1$ and $W_2$ are subspaces of $V$ such that $W_1 \cap W_2=\{0\}$ and $W_1 + W_2 = V$. We denote ...
0
votes
2answers
144 views

Proof of finitely dimensional subspaces

Prove: If $V_1, V_2, ...$ are finite dimensional subspaces of a Vector Space $V$, then for $n = 1, 2, ...$ $V_1 +...+ V_n$ is a finite dimensional subspace of V. I have the base case I assume true ...
1
vote
1answer
336 views

Basic arc length integration problem

How would you find the arc length of $r(t) = \langle\sqrt{t}, t,t^2\rangle$ for $1\le t\le 4$? This isn't a homework question, I'm just trying to understand how to properly solve a question such as ...
1
vote
5answers
192 views

Is closure of convex subset of $X$ is again a convex subset of $X$?

Today I was going through my text book exercise and got hold of the following question. Let $X$ be a normed linear space with norm $\lVert\cdot\rVert$ and $A$ is a non empty convex subset of $X$ ...
2
votes
2answers
893 views

Is closure of linear subspace of X is again a linear subspace of X??

Let $X$ be a normed linear space with norm $||\cdot||$ and $A \neq \emptyset$ is a linear subspace of $X$. Prove that $\bar{A}$ is also a linear subspace of $X$.
1
vote
2answers
1k views

Solid angle between vectors in n-dimensional space

There is a formula of to calculate the angle between two normalized vectors: $$\alpha=\arccos \frac {\vec{a} \cdot\ \vec{b}} {|\vec {a}||\vec {b}|}.$$ The formula of 3D solid angle between three ...
1
vote
1answer
89 views

Question about piecewise linear paths

I'm having trouble with a concept about piecewise linear paths. I've been looking on the net but I haven't found a definition of it. Consider a finite family of hyperplanes in a finite-dimension real ...
1
vote
3answers
91 views

Norm in $L^2$ space

I have a brief question regarding the norm of the $L^2$ space defined on an interval $[a,b]$. On various websites I have seen this defined as: $$\|f(x)\| = \int_{a}^{b} f(x)^2 dx$$ However, ...
2
votes
1answer
102 views

Finding a coset

I'm given $V$ a vector space over a field $\mathbb{F}$. Letting $v_1$ and $v_2$ be distinct elements of $V$, define the set $L\subseteq V$: $L=\{rv_1+sv_2 | r,s\in \mathbb{F}, r+s=1\}$. (This is the ...
1
vote
1answer
760 views

Solutions of Linear Homogenous Differential Equations as a Vector Space

If I'm not mistaken, the set of all functions $f(x)$ satisfying the first order homogeneous ODE: $$f''(x) - 2x = 0$$ is a Vector Space (as in, the elements of the Vector Space are its solutions). ...
13
votes
3answers
484 views

Pathologies in module theory

Linear algebra is a very well-behaved part of mathematics. Soon after you have mastered the basics you got a good feeling for what kind of statements should be true -- even if you are not familiar ...
3
votes
2answers
623 views

What is the difference between an array and a vector?

Okay so I'm doing a little bit of vector calculus at university (mainly with neural networks and the ...
3
votes
1answer
65 views

Finding some isomorphisms

Letting $U, V$ be vector spaces over $\mathbb{F}$ with $W\subseteq V$ a subspace. I want to show that if $B = \{T\in Hom_\mathbb{F}(U,V) | im(T)\subseteq W\}$ that $$B\approx ...
2
votes
4answers
93 views

Should I use sets or tuples when dealing with linear dependence?

Let set of vectors $\{x,y,z\}$ be linearly independent. Then would $\{x,y,z,x\}=\{x,y,z\}$ be linearly dependent, also? If so, that seems like a problem (since $\alpha x+\beta y+\gamma ...
4
votes
4answers
890 views

Sphere tangent to a plane

Find the equation for a sphere with center $(\alpha,\beta,\gamma)$ tangent to the plane $ax + by + cz = d$. The sphere is $(x-\alpha)^2 + (y-\beta)^2 +(z-\gamma)^2 = r^2$ and I understand that some ...
3
votes
3answers
1k views

Question on finite Vector Spaces, injective, surjective and if $V$ is not finite

Let $V$ be a vector space and $\alpha \in \operatorname{End}(V)$ (i) If $V$ is finite dimensional, then $\alpha$ is injective iff $\alpha$ is surjective. (ii) Give example showing (i) is false if ...
0
votes
2answers
77 views

Orthogonal vectors. Where am I going wrong?

I'm trying to show that given a set $\{\mathbf{a}, \mathbf{b}\}$ of orthonormal vectors in a 2-dimensional vector space, I can construct the identity matrix by computing $aa^\dagger + bb^\dagger$. ...
6
votes
3answers
422 views

Is this subset a subspace?

$$S = \{(x_1, x_2, x_3) \in \mathbb{R}^3\ |\ x_2 − (x_1)^2 = 0\}$$ I found this question in an old exam and I'm not sure how to prove this question, but I know it is not a subspace. Any help is ...
1
vote
1answer
423 views

What are the smallest and largest values of vectors

Let v and w be two vectors with ||v|| = 3 and ||w|| = 4. What is the largest and smallest possible values of v · w? I found this question in the textbook and I'm not sure how to solve this since I ...
1
vote
1answer
38 views

Replacing one of the conditions of a norm

Consider the definition of a norm on a real vector space X. I want to show that replacing the condition $\|x\| = 0 \Leftrightarrow x = 0\quad$ with $\quad\|x\| = 0 \Rightarrow x = 0$ does not alter ...
2
votes
1answer
94 views

Two questions about subspaces

I'm going through my assignments for this week, and I have a problem understanding the (notation of?) this exercise: Let $S$ be a nonempty set and $F$ a field. Prove that for any $s_0 \in S$, $\{f ...
3
votes
4answers
2k views

Showing vectors span a vector space by definition

I need to show that the vectors $v_1 = \langle 2, 1\rangle$ and $v_2 = \langle 4, 3\rangle$ span $\mathbb R^2$ by definition. By definition if I can write any vector in $\mathbb R^2$ as a linear ...
3
votes
1answer
68 views

Is the vector in the space of 3 other vectors

I have a set of 3 vectors $$ IE = {[1, 1, -3]; [2, -1, 3]; [-6, 3, -9]}$$ I want to know if the vector [1, 4, -12] , belongs (or is in the span?) to my previous set. So here's what I did. $$ ...
4
votes
2answers
2k views

Vector space of polynomials

Do all polynomials $ax^3 + bx^2 + cx + d$ with a root at $x=1$ form a vector space? Do the coefficients $(a,b,c,d)$ form a vector space? My reasoning: Since $x=1$ is a root, we can't have $(a,b,c,d)$ ...
0
votes
1answer
87 views

inequality between entries of the vector and $l_2$ norm of the vector

Let $a=(a_1, \ldots, a_n)$ be a vector in $R^n$. I am wondering for which vectors the following would be true: $$ \|a\|_2^2\geq c \sum_{i\ne j,i,j}a_ia_j, \quad i,j=1, \ldots, n. $$ Here $c>0$ is ...
0
votes
1answer
73 views

particle with radius 'r' hits the plane. what is the point of contact?

A large particle with radius r hits the plane with perpendicular 'n' and passing through ,q'. I need to check whether it hits the plane or not. In order to do that I need to find the point of contact. ...
2
votes
1answer
177 views

Directed line segments

The above question got me thinking how to resolve this AFTER ofcourse substituting the values.. tks
1
vote
1answer
155 views

Create wind animation

I'm trying to visually illustrate forecast wind speed and direction, the programming is the easy part, the math, I'm fuzzy on. I have a grid of points (lat/lon) , the forecast wind speed and ...
0
votes
1answer
537 views

vector space and general solution to the differential equation

The set of solutions of $(E): y' + a(x)y = 0$ ($a\,:\mathbb{R}\,\rightarrow\,\mathbb{R}$ continuous function) is a one-dimensional vector space. If $f(x) = e^{-\int_0^x a(t)\,\mathrm{d}t}$ is ...
0
votes
1answer
193 views

Infinite Direct Sums

I was told that every vector space can be written as a directsum of fields. However I don't see how this could be true for the space of all functions $f:\mathbb{R}\to\mathbb{R}$ (with addition and ...
10
votes
3answers
289 views

square root of $1/2 + \sqrt3/2?$

Playing with Maple, I noticed that it gives the square root of $c = 1+\frac{\sqrt3}{2}$ as equal to $a = \frac{1}{2}+\frac{\sqrt3}{2}$. Indeed it checks out. But I got curious: how can I find that ...
1
vote
2answers
73 views

Simple Vector Question in $\mathbb{R}^3$

Two points $A$ and $B$ in $\mathbb{R}^3$ with origin $O$ are given in terms of a Cartesian coordinate system by $A = (1, 2, 3)$ and $B = (4, 5, −1)$. How do you find the point $C$, such that $OACB$ ...
1
vote
1answer
212 views

Linear Algebra basic notation question

My book writes: A vector in $F^n$ may be regarded as a matrix $M_{n\times 1}(F)$. (true / false) What is $F$ or $F^n$, and how does the notation $M_{m\times n}(F)$ work? The books also likes to use ...
1
vote
1answer
112 views

Linear Algebra Basics

I'm having my first linear algebra classes in college right now, and a few difficulties with the symbolism used. Missing some basics so to say. So I have a few small questions I will just ask here: ...
2
votes
0answers
44 views

Randomized Solution to a System of Inequalities

Given a set of $\mathbf v_i \in \{0,1\}^k$ for $i=1,\dots,n$ and a vector $\mathbf x \in [0,1]^k$, we want to decide if the following inequality holds or not: $$ \mathbf x \le \sum_{i=1}^n \alpha_i ...
0
votes
1answer
45 views

The signage of a displacement vector?

Well, on my homework I had these two questions, but got a bit confused about what they are asking, and whether my answer is right or not. What is the sign of the displacement if you are moving ...
0
votes
2answers
67 views

Addition of a magnitude of a vector [closed]

The magnitude of vector a is 27, and the magnitude of vector b is 30. The direction, in degrees, for the vector a is 140, and the direction, in degrees, for the vector b is 69. What is the magnitude, ...
0
votes
2answers
2k views

Find initial velocity given initial speed and rate of deaccelration?

Okay, So we are going over vectors in class, given Cartesian coordinates and convert them to polar, and vice-versa. So the question is that a skateboard rolls up a ramp shown in the image shown, at a ...
3
votes
0answers
42 views

Symmetrizing a sequence of vectors

Given a finite set of real numbers $X_1, \ldots, X_n$, we can compute the first $n$ power sums of these numbers. From the power sums, the set $\{X_1, \ldots, X_n\}$ can be recovered. Essentially we ...