0
votes
1answer
18 views

Understanding the operator of differentiation on the vector space of polynomials

I have been reading through Linear Algebra Done Right by Sheldon Axler. The book defines an operator as a linear map from a vector space to itself. It then considers at another part of the book the ...
1
vote
2answers
40 views

What is wrong with this proof that if $V = U_1 \oplus W$ and if $V = U_2 \oplus W$, then $U_1 = U_2$?

Claim: Let $U_1, U_2$ and $W$ be subspaces of a vector space $V$. Suppose $V = U_1 \oplus W$ and $V = U_2 \oplus W$. Then $U_1 = U_2$. "Proof" Let $v \in V$. Then $\exists \space u_1 \in U_1 $ ...
2
votes
1answer
38 views

Problems on vector spaces

Let $E$ a $\mathbb{K}$-vector space of finite dimension $n$, $\mathcal{V}$ a subspace of $\mathcal{L}(E)$ such that $$\forall u\in\mathcal{V}\setminus \{0\},u\in\mathcal{GL}(E)$$ a) Show that ...
0
votes
2answers
22 views

Solution to homogeneous linear differential equation form a vector space

Show that the solutions of a homogeneous linear differential equation $y"+a(x)y'+b(x)y = 0$ form a vector space. What is its dimension? I understand that the dimension is 2 and that 0 is a solution ...
2
votes
1answer
16 views

Computing intersection of vector spaces spanned by two lists

Assume that I'm given two lists of vectors $l_1$ and $l_2$, where all the vectors have equal dimension. I want to compute a basis for the intersection of their spans. What is the easiest setup for ...
0
votes
1answer
29 views

Subsets that are also vector spaces

The vector space $R^3$ and the subset M consists of the vectors $(\xi_1,\xi_2,\xi_3)$ for which i) $\xi_1 = 0 $ ii) $\xi_1 = 0$ or $\xi_2 = 0 $ iii) $\xi_1 + \xi_2 = 0 $ iv) $\xi_1 + \xi_2 = 1 $ ...
0
votes
1answer
22 views

Function from one Null space to Another

Suppose a single vector space over $R$ of degree $n$, and two matrices $A, B$ of arbitrary row size, but col size $n$, s.t. their individual null spaces are linear subspaces of this vector space. Is ...
1
vote
1answer
20 views

For a linear function, the fiber of the output is the translate of the kernel by the input. (Trivial observation, proof needed.)

As you may already know, I am a newbie to linear algebra. I am supposed to prove that for every linear function between vector spaces, for every input, the fiber of the corresponding output equals the ...
0
votes
1answer
29 views

Linear Algebra Vector Tracing

Let $A(2,-1,1)$, $B$ and $C$ be the vertices of a triangle where $\overrightarrow{AB}$ is parallel to $\vec{v}=(2,0,-1), $$\overrightarrow{BC}$ is parallel to $\vec{w}=(1,-1,1)$ and $\angle(BAC)=90°$. ...
1
vote
1answer
36 views

Question about dimension of a subspace

Let $K$ be a field and define the following subspaces $$V=\textrm{span}(e_1,e_2,e_3),\;\; V^\bot = \textrm{span}(e_4,e_5,e_6)$$ inside $K^6$. Let $\dim L=4$ and assume that $\dim L\cap V\leq 1$. Can ...
1
vote
0answers
43 views

Basis of $\mathbf{Q}[x]$

I wanna show that the binomials $\binom{x}{k}$ for $k=0,1,\ldots$ form a basis of the $\mathbf{Q}$-vector space $V=\mathbf{Q}[x]$. I can show that for fixed $m\in\mathbf{N}$ the $\binom{x}{k}$ ...
1
vote
0answers
13 views

Use of Matlab to put equation into vector form

Is there a way to put the following equation of a line into vector form using Matlab? $\displaystyle y=\frac{cos(s_n)-cos(s_{n+1})}{sin(s_{n+1}-sin(s_n)}(x-sin(s_n))-cos(s_n)$
1
vote
1answer
24 views

Deduction of vector form of Snell's law

I was unable to find the deduction of the vector form of Snells's law. $$n_1\sin\theta_1 = n_2\sin\theta_2$$ Here is the vector form, from the article A Theory of Multi-Layer Flat Refractve Geometry ...
1
vote
0answers
28 views

linear algebra question

Consider $n$ convex polytopes $S_1, \cdots, S_n$ and a set of matrices $W$ such that each matrix $A\in W$, we have that the $i$-th row of $A$ is a member of $S_i$. (In general $W$ is infinite.) ...
0
votes
1answer
34 views

$dim_\mathbb C V=n$ then $dim _\mathbb R V=2n$

Prove that if the dimension of a vector space $V$ over $\mathbb C$ is $n$ then the dimension of $V$ over $R$ is $2n$ I wanted to do it using isomorphisms i.e. every finite dimensional vector space ...
0
votes
0answers
18 views

How to prove for an operator $L$ on a vector space $V$ that $Null(L^k)\subset$ $Null(L^{k+1})$?

This was a past exercise and I still struggle to understand why it is necessary to prove it. I could very well be doing it wrong too! We have $L$, an operator (I'm assuming linear, but feel free to ...
0
votes
1answer
17 views

A Quotient space Problem

Prove that there is a natural isomorphism between $(V/W)'$ and $W^0$ where $W^0$ is the annihilator of $W$ and $(V/W)'$ is the dual of $V/W$
0
votes
2answers
52 views

Prove that $L(V,W)$ forms a vector space

Let $V$ and $W$ be vector spaces over a field $F$. Let $L(V,W) = \{T:V\to W : T \text{ is linear} \}$, that is, $L(V,W)$ is the collection of all linear functions from $V$ to $W$. For $S,T \in L(V,W)$ ...
2
votes
1answer
36 views

Hamel Basis Exercise Proof Clarification.

While looking up something else on stack exchange, I ran across this question An exercise about a Hamel basis and it intrigued me. The answer was provided by Jonathan Golan ...
0
votes
0answers
38 views

Linear Algebra (Basis)

We have B: Question: Find a basis in $\mathbb{M}_{3,2}(\mathbb{R}$) that has B. Obs.: I have no idea how to do this. I know that a combination of a basis is a vector in the subspace formed for this ...
2
votes
0answers
17 views

Linear algebra (Coordinates)

Question: Find the coordinates of $x=(1,0,0)$ in relation to base $$B=\{(1,1,1),(-1,1,0),(1,0,-1)\}.$$ I tried: $a,b,c\in R$ such that $$a(1,1,1)+b(-1,1,0)+c(1,0,-1)=(1,0,0)=x$$ but I'm not sure ...
0
votes
1answer
21 views

Is this an Alternative Proof of a set of vectors forming a basis?

This is one of my exam past paper question So I proved this correctly by following the normal method which is showing that a, b and c are linearly independant My proof - When I looked at the ...
1
vote
1answer
30 views

Find a basis of $M_2(F)$ so that every member of the basis is idempotent

Let $V=M_{2\times 2}(F)$ (the space of 2x2 matrices with coefficients in a field $F$). Find a basis $\{A_1,A_2,A_3,A_4\}$ of $V$ so that $A_j^2=A_j$ for all $j$. My attempt. Let $A_j$ be ...
0
votes
1answer
36 views

Nonhomogeneous Linear Systems and Vector Space Solutions

Are there any nonhomogeneous linear systems with a solution set that forms a vector space? I know that, in order to be a vector space, a set must consists of a set V together with operations + (called ...
0
votes
1answer
33 views

being $\mathbf{w}$ a vector, how do I calculate the derivative of $\mathbf{w}^T\mathbf{w}$?

Let's say that I have a vector $\mathbf{w}$. How can I calculate the derivative in the following expression? $\frac{\mathrm{d}}{\mathrm{d}\mathbf{w}}\mathbf{w}^T\mathbf{w}$
0
votes
0answers
25 views

being $\mathbf{a}$ and $\mathbf{b}$ two vectors with same length, how do I expand $(\mathbf{a}^T\mathbf{b})^2$?

Let's say that I have two vectors $\mathbf{a}$ and $\mathbf{b}$. Assuming that they have same length, their product $\mathbf{a}^T\mathbf{b}$ and its square $(\mathbf{a}^T\mathbf{b})^2$ are scalars. ...
0
votes
1answer
24 views

Proof linear independency lemma

If $\mathbf{u}$ and $\mathbf{v}$ is in the complex vector space $V$ and $\mathbf{w}_1 = \mathbf{u} + i \mathbf{v}$ and $\mathbf{w}_2 = \mathbf{u} - i \mathbf{v}$ are linear independent then will the ...
0
votes
1answer
29 views

Transformation from cartesian to polar Coordinates of Vector Field

This is fairly low-level, still I would like to get a verification: I vector field $$\mathbf{F}=F_x \hat{e_x} + F_y \hat{e_y} = F_r \hat{e_r} + F_{\phi} \hat{e_\phi}$$ given in cartesian coordinates, ...
0
votes
0answers
29 views

Relation between basis elements under automorphisms

Let $e_1, e_2, e_3$ denote the standard basis for the vector space $\mathbb{R}^3$, and let $f, g: \mathbb{R}^3\to \mathbb{R}^3$ be linear maps such that $g\circ f = f\circ g = {\rm id}$. Also, let ...
2
votes
0answers
115 views
+50

Vector space basis change: is this “index-free” notation correct?

There are already quite a number of questions on basis change in a vector space. Nevertheless, to fully grasp the underlying idea I made up the following notation and I have some doubts on it (note: ...
0
votes
0answers
19 views

A question on existence of a linear map on finite dimensional vector spaces

Suppose that $V, W$ are finite dimensional vector spaces and $U$ a subspace of $V$ such that $\dim U\ge \dim V-\dim W$ , then how do we prove that there is a linear map $T:V\to W$ such that $\ker ...
2
votes
1answer
41 views

Finding set of vectors that spans the solution set

Question: Find a set of vectors $\{u,v\}$ in $\mathbb{R}^4$ that spans the solution set of the equations: $$\begin{align}w - x + y + z = 0 \\ 5w + 2x - y + z = 0\end{align}$$ Reducing these I get: ...
0
votes
0answers
36 views

normal vectors in spaces where $n > 3$

I am reading Lovelock and Rund's book on Tensors and they make a statement that I wanted to validate about normal vectors in high-dimensional spaces. It should be remarked that the above ...
0
votes
0answers
30 views

Proving basis of $\mathcal{L}(V,W)$

Suppose two vector spaces $V$ and $W$ over some field $F$ is given. Now let $\mathcal{L}(V,W)$ be the set of all linear maps from $V$ to $W$. Also let $\dim V=n,\dim W=m$ and $\mathscr{M}(m,n,F)$ be ...
0
votes
2answers
56 views

General vector space theory developed without matrix-theory.

Since vector spaces can exist regardless of a matrix I wanted to see if we could do all the proofs for the general vector-spaces without using theory for matrices. Then it was only two proofs of the ...
4
votes
1answer
47 views

Does $(x,f(x),\cdots,f^p(x))$ is linearly dependent over $E$ implies $(id, f, …, f ^ p)$ is linearly dependent over $\mathcal{L}(E)$?

Here is the original (classic I think) problem I had encored: if $(x,f(x))$ is a linearly dependent family of $E$ (a vector space) for all $x\in E$, then the family $(id,f)$ is linearly dependentt ...
0
votes
0answers
30 views

Prove that if $m<n$ then $S$ does not generate $V$

Let $V$ be a vector space over a field $K$ such that $dim V=n$ and let $S\subseteq V$ such that $|S|=m$. Prove that if $m<n$ then $S$ does not generate $V$ Let $S=${$s_1,...,s_m$}. Suppose that ...
1
vote
1answer
23 views

definition of linearly dependent set

I know that this is a silly question to ask but I would really appreciate if you can answer me. Let $V$ be a vector space and $S=\{v_1,\ldots,v_n\}$ a finite subset of $V$. $S$ is linearly dependent ...
3
votes
2answers
54 views

A question on dual spaces of vector spaces

I've been doing a bit of self study into the formalism of dual spaces. In the book that I've been reading the author introduces the notion of a dual space, $V^{\ast}$ to a given vector space $V$ as ...
0
votes
1answer
25 views

Linear algebra perpendicular vectors

How do I know the vectors that are perpendicular to $(1,1,1)$ and $(1,2,3)$ lie on a line?, thanks beforehand, I'll appreciate any help here.
0
votes
1answer
20 views

Sub-space problem in complex space

A problem in my professor's guide is driving me nuts, I don't even know where to start, this is the problem: Is $ \{(z,u) \in \mathbb{C}^2 / z - \overline{z} + u = 0\} $ a sub-space of $ ...
1
vote
2answers
31 views

Rank of $I_m - X_{m \times m}$ given rank of $X$

I have a matrix $X_{m\times m}$ which is idempotent and has $rank(X) = n < m$. I have for some time now been trying to calculate $rank(I_m - X)$ but have been unable to do so. I should be able to ...
0
votes
1answer
29 views

Linear operators and change of basis in a vector space

Suppose we have a vector space $V$ over a scalar field $\mathbb{F}$ and two different bases $\mathcal{B}=\lbrace\mathbf{v}_{i}\rbrace_{i=1,\ldots , n}$ and ...
1
vote
4answers
24 views

Let $X,Y,Z$ be subspaces of $V$ so that $X$ is a subspace of $Y$. Prove that $Y\cap (X+Z)=X+(Y\cap Z)$

Let $X,Y,Z$ be subspaces of $V$ so that $X$ is a subspace of $Y$. Prove that $Y\cap (X+Z)=X+(Y\cap Z)$ I know that I need to prove that $Y\cap (X+Z)\subseteq X+(Y\cap Z)$ and $X+(Y\cap Z)\subseteq ...
0
votes
2answers
34 views

What does it mean when dim(V)=rankT

I have a question relating to a linear transformation and have ended up with the result that $dim(V)=rank(T)$. I got to this because I'm told that $V$ and $W$ are finite dimensional vector spaces, ...
1
vote
3answers
27 views

Prove that $\exists y\in V$ so that the set {${u+y:u\in U}$} is a subspace of $V$

Let $V$ a vector space over a field $F$, and let $v,w\in V$ so that $v\neq w$. Define $U=${${(1-t)v+tw: t\in F}$}. Prove that $\exists y\in V$ so that the set {${u+y:u\in U}$} is a subspace of $V$ ...
0
votes
0answers
77 views

Proof of the cardinality of continuous functions from $[0,1]$ to $[0,1]$.

I've been thinking about the cardinality of continuous functions from $[0,1]$ to $[0,1]$. I know that the cardinality is the same as that of $[0,1]$ and the standard proof using the fact that such a ...
1
vote
0answers
54 views

Why does the law of cosine not work in $\mathbb R^n$?

In class, we derived the dot product formula using the cosine law in $\mathbb R^2$ and $\mathbb R^3$. Then, we found the Cauchy-Schwarz theorem and the Triangle Inequality theorem. Then, we defined ...
0
votes
1answer
17 views

Define two operations in $V$={${i\in \mathbb N: i<2^n}$}

Let $V$={${i\in \mathbb N: i<2^n}$}, with n fixed. Define an operation for addition and an operation for scalar multiplication so that $V$ with those operations is a vector space over $\mathbb Z_2$ ...
1
vote
0answers
37 views

Variation of orthogonal vectors

It is given that inner product $$ \left\langle a(t),b(t)\right\rangle =0,\quad \forall t\in[0,T] $$ where $a(t), b(t)\in \mathbb{R}^n$. If $\dot{a}(t)$ is known, is there a way to find an expression ...