Vector space Definition and 530 Threads

Consider the rank-nullity theorem. We want to prove that for a linear transformation ##\mathsf T:\mathsf V\to\mathsf W##, $$\operatorname{nullity}(\mathsf T)+\operatorname{rank}(\mathsf T)=\operatorname{dim}(\mathsf V).$$We have a basis ##\{v_1,\ldots,v_k\}## of the null space ##\mathsf...

In Introduction to Topology by Gamelin and Greene, I'm working an exercise to show the equivalence of norms in ##\mathbb R^n##. This exercise succeeds another exercise where various equivalent formulations of "equivalent norms" have been given, e.g. that two norms ##\|\cdot\|_a,\|\cdot\|_b## are...

Hi, a doubt about the definition of vector space. Take for instance the set of polynomals defined on a field ##\mathbb R ## or ##\mathbb C##. One can define the sum of them and the product for a scalar, and check the axioms of vector space are actually fullfilled. Now the point is: if one...

In the book I'm reading, Before Machine Learning, by Jorge Brasil, I'm on the section that introduces bases for vector spaces. The author gives the example of a vector space with two vectors ##\vec i## and ##\vec j## forming the basis where ##\vec i = (1,0)## and ##\vec j = (0,1)## He then says...

Hi, from Penrose book "The Road to Reality" it seems to me inner product and dot/scalar product are actually different things. Given a vector space ##V## an inner product ## \langle . | . \rangle## is defined between elements (i.e. vectors) of the vector space ##V## itself. Differently...

Here is one proof $$\forall u\in U\implies Tu\in U\subset V\implies T^2u\in U\implies \forall m\in\mathbb{N}, T^m\in U\tag{1}$$ Is the statement above actually a proof that ##\forall m\in\mathbb{N}, T^m\in U## or is it just shorthand for "this can be proved by induction"? In other words, for...

Determine whether the following subsets U of M4x4is a subspace of the vector space V of all M4x4 matrices, with the standard operations of matrix addition and scalar multiplication. If is not a subspace provide an example to demonstrate a property that U does not possess. a. The set U of all 4x4...

Dear Everybody, I am having trouble with last part of this question. I believe the answer is no. But I have to proof the general case. Here is my work for the problem: Suppose that we have two distinct norms on the same vector space ##X## over complex numbers. Then there exists no ##K## in...

I don't really know how I am supposed to approach that. In general, I know how to show that a function is linear, which is to show that ##f(\alpha \cdot x) = \alpha \cdot f(x)## and ##f(x_1 + x_2) = f(x_1) + f(x_2)##. However, for this specific function, I have no idea, since there is nothing...

Suppose M is a manifold and $$T_{p}M$$ is the tangent space at a point $$p \in M$$. How do i prove that it is indeed a vector space using the axioms: Suppose that u,v, w $$\in V$$. where u,v, w are vectors and $$\V$$ is a vector space $$u + v \in V \tag{Closure under addition}$$ $$u + v = v +...

Let C2x2 be the complex vector space of 2x2 matrices with complex entries. Let and let T be the linear operator onC2x2 defined by T(A) = BA. What is the rank of T? Can you describe T2? ____________________________________________________________ An ordered basis for C2x2 is: I don't...

##f : [0,2] \to R##. ##f## is continuous and is defined as follows: $$ f = ax^2 + bx ~~~~\text{ if x belongs to [0,1]}$$ $$ f(x)= Ax^3 + Bx^2 + Cx +D ~~~~\text{if x belongs to [1,2]}$$ ##V = \text{space of all such f}## What would the basis for V? Well, for ##x \in [0,1]## the basis for ##V##...

Let ##S## be the subset of real (infinite) sequences (##a_1,a_2,\ldots##) with ##\lim a_n=0## and let ##V## be the space of all real sequences. Is ##S## a subspace of ##V##? Hello. I want to ask for help to start solving this problem. I don't understand how I can apply the theory I've studied...

I'm learning Linear Algebra by self and I began with Apsotol's Calculus Vol 2. Things were going fine but in exercise 1.13 there appeared too many questions requiring a strong knowledge of Real Analysis. Here is one of it (question no. 14) Let ##V## be the set of all real functions ##f##...

Is the following a correct demonstration that quantum mechanics can be done in a real vector space? If you simply stack the entries of density matrices into a column vector, then the expression ##\textrm{Tr}(AB^\dagger)## is the same as the dot product in a complex vector space (Frobenius inner...

Could someone explain the green highlight to me, please?

Hi PF, I've one question about vector spaces. There is only one way to define the operations of a vector space? For example if V is a vector space there is other way to define their operations like scalar multiplication or the sums of their elements and that the result is also a vector space?

I read this in the wiki article about Wick rotation: Note, however, that the Wick rotation cannot be viewed as a rotation on a complex vector space that is equipped with the conventional norm and metric induced by the inner product, as in this case the rotation would cancel out and have no...

So the reason why I'm struggling with both of the problems is because I find vector spaces and subspaces hard to understand. I have read a lot, but I'm still confussed about these tasks. 1. So for problem 1, I can first tell you what I know about subspaces. I understand that a subspace is a...

Solution 1. Based on my analysis, elements of ##V## is a map from the set of numbers ##\{1, 2, ..., n\}## to some say, real number (assuming ##F = \mathbb{R}##), so that an example element of ##F## is ##x(1)##. An example element of the vector space ##F^n## is ##(x_1, x_2, ..., x_n)##. From...

Be ##T_{1}, T_{2}## upper and lower matrix, respectivelly. Show that we haven't matrix ##M(NxN)## such that ##M(NxN) = T_{1}\bigoplus T_{2}## I am not sure if i get what the statement is talking about, can't we call ##T_{1},T_{2} = 0##? Where 0 is the matrix (NxN) with zeros on all its entries...

Summary:: Problem interpreting a vector space of functions f such that f: S={1} -> R Hello, Another question related to Jim Hefferon' Linear Algebra free book. Before explaining what I don't understand, here is the problem : I have trouble understanding how the dimension of resulting space...

Summary:: Be the set X of vectors {x1,...,xn} belong to the vector space E. If this set X is convex, prove that all the convex combination of X yet belong to X. Where convex combination are the expression t1*x1 + t2*x2 + ... + tn*xn where t1,...,tn >= 0 and t1 + ... + tn = 1 I tried to suppose...

b) c and d): In c) I say that ##L_h## is only self adjoint if the imaginary part of h is 0, is this correct? e) Here I could only come up with eigenvalues when h is some constant say C, then C is an eigenvalue. But I' can't find two.Otherwise does b-d above look correct? Thanks in advance!

How can I prove that Zp is a vector space if and only if p is prime

Hello, I am doing a vector space exercise involving functions using the free linear algebra book from Jim Hefferon (available for free at http://joshua.smcvt.edu/linearalgebra/book.pdf) and I have trouble with the author's solution for problem II.1.24 (a) of page 117, which goes like this ...

If I'm given a set of four vectors, such as A={(0,1,4,2),(1,0,0,1)...} and am given another set B, whose vectors are given as a form such as (x, y, z, x+y-z) all in ℝ, what steps are needed to show A is a basis of B? I have calculated another basis of B, and found I can use linear combinations...

Hi I believe I understand the concept of a vector space V and its dual V*. I also understand that for V finite dimensional, there is a natural isomorphism between V and V**. What I am struggling to understand is - Does this natural isomorphism mean that V** is always IDENTICAL to V (identical...

Good Morning Recently, I asked why there must be two possible solutions to a second order differential equation. I was very happy with the discussion and learned a lot -- thank you. In it, someone wrote: " It is a theorem in mathematics that the set of all functions that are solutions of a...

This exercise is located in the vector space chapter of my book that's why I am posting it here. Recently started with this kind of exercise, proof like exercises and I am a little bit lost Proof that given a, b, c real numbers, the set X = {(x, y) E R^2; ax + by <= c} ´is convex at R^2 the...

Summary:: the set of arrays of real numbers (a11, a21, a12, a22), addition and scalar multiplication defined by ; determine whether the set is a vector space; associative law Question: determine whether the set is a vector space. The answer in the solution books I found online says that...

I understand that the Dual Space is composed of elements that linearly map the elements of the Vector Space onto Real numbers If my preamble shows that I have understood correctly the basic premise, I have one or two questions that I am trying to work through. So: 1: Is there a one to one...

As per source # 1 ( link below), when treating polynomials as vectors, we use their coefficients as vector elements, similar to what we do when we create matrices to represent simultaneous equations. However, what I noticed in Source #2 was that, when functions are represented as vectors, the...

I am assuming the set ##V## will have elements like the ones shown below. ## v_{1} = (200, 700, 2) ## ## v_{2} = (250, 800, 3) ## ... 1. What will be the vector space in this situation? 2. Would a subspace mean a subset of V with three or more bathrooms?

If one shows that ##U\cap V=\{\textbf{0}\}##, which is easily shown, would that also imply ##\mathbf{R}^3=U \bigoplus V##? Or does one need to show that ##\mathbf{R}^3=U+V##? If yes, how? By defining say ##x_1'=x_1+t,x_2'=x_2+t,x_3'=x_3+2t## and hence any ##\textbf{x}=(x_1',x_2',x_3') \in...

Given that the Set of 1-Forms is a Vector Space distinct from, but complimentary to, the Linear Vector Space of Vectors. And given that there is an Isomorphism between the linear space of vectors and the dual vector space of 1-forms, does it make mathematical sense to combine a vector space and...

--##ker(T^2)=ker(T)## if ##T(V)=T^2(V)##-- Suppose that ##T^2(V)=T(V)##. So ##T:T(V)\mapsto T^2(V)=T(V)##. Hence, ##T## is one-to-one and so ##ker(T)=\{0\}##. Suppose that ##T^2(w)=0## for some ##w\in ker(T^2)##. Then ##T^2(w)=T(T(w))=0## which implies that ##T(w)\in ker(T)## and so ##T(w)=0##...

I am reading N. L. Carothers' book: "Real Analysis". ... ... I am focused on Chapter 3: Metrics and Norms ... ... I need help Exercise 32 on page 46 ... ... Exercise 32 reads as follows: I have not been able to make much progress ... We have ...$$B_r(x) = \{ y \in M \ : \ d(x, y) \lt r...

The standard definition of the basis for a vector space is that all the vectors can be defined as finite linear combinations of basis elements. Consider the vector space consisting of all sequences of field elements. Basis vectors could be defined as vectors which are zero except for one term in...

Determine if the set of vectors $\begin{bmatrix} x\\y\\5 \end{bmatrix}\in \Bbb{R}^3$ form a vector space ok if I follow the book example I think this is what is done $\begin{bmatrix} x_1\\y_2\\5 \end{bmatrix} +\begin{bmatrix} x_2\\y_2\\5 \end{bmatrix} +\begin{bmatrix} x_2\\y_2\\5...

Homework Statement Problem given to me for an assignment in a math course. Haven't learned about roots of unity at all though. Finding this problem super tricky any help would be appreciated. Screenshot of problem below. [/B] Homework Equations Unsure of relevant equations The Attempt at...

I am reading the book: Multivariable Mathematics by Theodore Shifrin ... and am focused on Chapter 8, Section 2, Differential Forms ... I need some help in order to fully understand the vector space of alternating multilinear functions ... The relevant text from Shifrin reads as follows: In...

Hey! :o Let $S_{X,3}$ be the vector space of cubic spline functions on $[-1,1]$ in respect to the points $$X=\left \{x_0=-1, x_1=-\frac{1}{2}, x_2=0, x_3=\frac{1}{2}, x_4\right \}$$ I want to check if the function $$f(x)=\left ||x|^3-\left |x+\frac{1}{3}\right |^3\right |$$ is in $S_{X,3}$...

this is what is given so by addition $$\begin{bmatrix}x_1\\y_1\\5z_1\end{bmatrix} \oplus \begin{bmatrix} x_2\\y_2\\5z_2 \end{bmatrix} = \begin{bmatrix} x_1+x_2\\y_1+y_2\\5z_1+5z_2 \end{bmatrix} = \begin{bmatrix} X\\Y\\10Z \end{bmatrix}$$ uhmmmm really?

On the set of vectors $\begin{bmatrix} x_1 \\ y_1 \end{bmatrix}\in \Bbb{R}^2 $ with $x_1 \in \Bbb{R}$, and $y_1$ in $\Bbb{R}^{+}$ (meaning $y_1 >0$) define an addition by $$\begin{bmatrix} x_1 \\ y_1 \end{bmatrix} \oplus \begin{bmatrix} x_2 \\ y_2 \end{bmatrix} = \begin{bmatrix} x_1 + x_2 \\...

Homework Statement Show that the only subspaces of ##V = R^2## are the zero subspace, ##R^2## itself, and the lines through the origin. (Hint: Show that if W is a subspace of ##R^2## that contains two nonzero vectors lying along different lines through the origin, then W must be all of...

Given a basis of a vector space $(V,O_1,O_2)$ can it represent two different non-isomorphic graphs.Any other inputs kind help. It will improve my knowledge way of my thinking. Another kind help with this question is suppose (V,O_1,O_2) and (V,a_1,a_2) are two different vector spaces on the...

Hey! :o We consider the $\mathbb{F}_2$-vector space $(2^M, +, \cap)$, where $M$ is non-empty set and $+ : 2^M\times 2^M \rightarrow 2^M: (X,Y)\mapsto (X\cup Y)\setminus (X\cap Y)$. I want to show that $(2^M, +, \cap )$ for $\mathbb{K}=\{\emptyset , M\}$ satisfies the axioms of a vector space...

Hey! :o Let $1\leq n\in \mathbb{N}$ and let $U_1, U_2$ be subspaces of the $\mathbb{R}$-vector space $\mathbb{R}^n$. I want to prove or disprove the following: The set $\{f\in \mathbb{R}^{\mathbb{R}} \mid \exists x\in \mathbb{R} : f(x)=0_{\mathbb{R}}\}$ is a subspace of...

It is well known that the set of exponential functions ##f:\mathbb{R}\rightarrow \mathbb{R}_+ : f(x)=e^{-kx}##, with ##k\in\mathbb{R}## is linearly independent. So is the set of sine functions ##f:\mathbb{R}\rightarrow [-1,1]: f(x) = \sin kx##, with ##k\in\mathbb{R}_+##. What about...