Vector space Definition and 540 Threads

In the book that I read, an operator is defined to be a linear map which maps from a vector space into itself. For example, if ##T## is an operator in a vector space ##V##, then ##T:V\rightarrow V##. Now, what if I have an operator ##O## such that ##T:V\rightarrow U## where ##U## is a subspace...

Two examples are: Given two subspaces ##U## and ##W## in a vector space ##V##, the smallest subspace in ##V## containing those two subspaces mentioned in the beginning is the sum between them, ##U+W##. The smallest subspace containing vectors in the list ##(u_1,u_2,...,u_n)## is...

I am reading Bruce Cooperstein's book: Advanced Linear Algebra ... ... I am focused on Section 1.3 Vector Spaces over an Arbitrary Field ... I need help with Exercise 1 of Section 1.3 ... Exercise 1 reads as follows:Although apparently simple, I cannot solve this one and would appreciate...

Homework Statement Find basis and dimension of V,W,V\cap W,V+W where V=\{p\in\mathbb{R_4}(x):p^{'}(0) \wedge p(1)=p(0)=p(-1)\},W=\{p\in\mathbb{R_4}(x):p(1)=0\} Homework Equations -Vector spaces The Attempt at a Solution Could someone give a hint how to get general representation of a vector...

Homework Statement Find a basis for the following vector space: ## V = \{ p \in \mathbb C_{\leq4} ^{[z]} | \ p(1)=p(i) ## and ## p(2)=0 \} ## (Where ## \mathbb C_{\leq4} ^{[z]} ## denotes the polynomials of degree at most 4) Homework Equations N/A The Attempt at a Solution I tried to find...

Homework Statement First, I'd like to say that this question is from an Introductory Linear Algebra course so my knowledge of vector space and subspace is limited. Now onto the question. Q: Which of the following are subspaces of F(-∞,∞)? (a) All functions f in F(-∞,∞) for which f(0) = 0...

Homework Statement Let |v> ∈ V with |v> ≠ |0>, and let λ, μ ∈ ℂ. Prove that if λ|v> = μ|v>, then λ = μ Homework Equations Vector Space Axioms The Attempt at a Solution I am struggling to begin with this one. I can think of tons of different ways to begin, but all seem to get into a hazy area...

Homework Statement Let v ∈ V and c ∈ ℂ, with c ≠ 0. Prove that if cv = 0, then v = 0. Homework Equations Vector space axioms. The Attempt at a Solution Simple proof overall, but I have one major clarification question. v = 1v = (c^(-1)c)v = c^(-1) (cv) = c^(-1) 0 v = 0 My question is, in...

Dear Physics Forum personnel, I am curious if the euclidean space R^n is an example of vector space. Also can matrices with 1x2 or 2x1 dimension be a vector for the R^n? PK

<< Thread moved to the HH forums from the technical engineering forums, so no HH Template is shown >> The model: The goal: 1. Create a three phase voltage 2. Do a alpha-beta transformation 3. Do a Cartesian to Polar transformation 4 Check the output angle The expected result: Since the space...

So when I took linear algebra I was asked to show the well-known properties of orthogonality in the functions cos(mx) and sin(mx)... With this I mean all the usual combinations which I don't see why I should point out explicitly. The inner product in the vector space was defined...

I may have a bad day, or not enough coffee yet. So, "If A is a nonempty subset of a vector space V, then the set L(A) of all linear combinations of the vectors in A is a subspace, and it is the smallest subspace of V which includes the set A. If A is infinite, we obviously can't use a single...

if we do picard's iteration of nth order linear ODE in the vector form, we can show that nth order linear ODE's solution exists. (5) (17) example) (21) (22) (http://ghebook.blogspot.ca/2011/10/differential-equation.html)I found that without n number of initial conditions, the solution...

Hi Folks, I find this link http://mathworld.wolfram.com/VectorSpaceBasis.html confusing regarding linear independence. One of the requirement for a basis of a vector space is that the vectors in a set S are linearly independent and so this implies that the vector cannot be written in terms of...

$\newcommand{\R}{\mathbf R}\newcommand{\C}{\mathbf C}$ Low-Tech Complexification: Let $V$ be a finite dimensional vector space over $\R$. We can forcefully make $W:=V\times V$ into a complex vector space by defining addition component-wise and product $\C\times W\to W$ as $$ (a+ib)(u...

What is the difference between a vector and a vector space? I get that a vector is an object with both magnitude and direction, but am confused by the term "vector space". Does a vector space simply refer to a collection of vectors? Thanks!

Homework Statement (a) Show that the set of all square-integrable functions is a vector space. Is the set of all normalised functions a vector space? (b) Show that the integral ##\int^{a}_{b} f(x)^{*} g(x) dx## satisfies the conditions for an inner product. Homework Equations The main...

Homework Statement Let V be a vector space over a field F and let L and M be two linear transformations from V to V. Show that the subset W := {x in V : L(x) = M(x)} is a subspace of V .The Attempt at a Solution I presume it's a simple question, but it's one of those where you just don't...

4b). How can I find a basis? I was thinking of the standard basis $\{1,x,x^2\}$, but that doesn't work under the scalar multiplication definition in the vector space. EDIT: I think it is $\{0,x,x^2\}$ and we take $1$ to be the $0$ vector! $a(0)+b(x)+c(x^2)=1$ implies $a=b=c=0$. It is strange...

Wiki says "A vector space is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars in this context." To me the term (linear) Vector Space has always seemed a little mysterious ... how far wrong would I be in thinking of a vector...

Homework Statement Hello, here is the question: "Rather than use the standard definitions of addition and scalar multiplication in R3, suppose these two operations are defined as follows. With these new definitions, is R3 a vector space? Justify your answers. a) (x1, y1, z1) + (x2, y2, z2) =...

Hi, i have been going through some elementray reading on group theory. if g(θ) is a group element parameterized by the continuous variable θ. g(θ i )| θ i =identity, if D(θ i ) is a matrix representation on a d dimensional vector space V . What is D(θ i )| θ i =0 ?

Homework Statement Let V be a vector space, and suppose that \vec{v_1}, \vec{v_2}, ... \vec{v_n} is a basis of V. Let c\in\mathbb R be a scalar, and define \vec{w} = \vec{v_1} + c\vec{v_2}. Prove that \vec{w}, \vec{v_2}, ... , \vec{v_n} is also a basis of V. Homework Equations If two of the...

Homework Statement Determine if they given set is a vector space using the indicated operations. Homework EquationsThe Attempt at a Solution Set {x: x E R} with operations x(+)y=xy and c(.)x=xc The (.) is the circle dot multiplication sign, and the (+) is the circle plus addition sign. I...

Homework Statement The set ℝ^2 with vector addiction forms an abelian group. a ∈ ℝ, x = \binom{p}{q} we put: a ⊗ x = \binom{ap}{0} ∈ ℝ^2; this defines scalar multiplication ℝ × ℝ^2 → ℝ^2 (p, x) → (p ⊗ x) of the field ℝ on ℝ^2. Determine which of the axioms defining a...

Homework Statement Let V be a vector space over the field F. The constant, a, is in F and vectors x, y in V. (a) Show that a(x - y) = ax - ay in V . (b) If ax = 0_V show that a = 0_F or x = 0_V . Homework Equations axiom 1: pv in V, if v in V and p in F. axiom 2: v + v' in V if v, v' in V...

Dear all, Can anyone please explain how the linear combination of non-coplanar and non-orthogonal coordinate axes representing a point x as shown below is derived. Please use the reference text attached in this post to explain to me as i will find it a bit relevant. I want to...

The thing is in my undergrad I haven't gone into a class that includes discussion about vector space, and the related stuffs, simply because they were not offered in the syllabus. As I have seen in some Quantum physics courses in some universities, they did talk about vector space in a certain...

So I was thinking and I was wondering if we could have a set of vectors that spanned just the unit sphere, and nothing else beyond that. So, if we replace euclid's 5th postulate to give us spherical geometry, a line is a circle on the surface of some sphere. If we have two perpendicular vectors...

Let $V$ be a vector space, and define $V^n$ to be the set of all n-tuples $(v_1, v_2,...,v_n)$ of n vectors $v_i$, each belonging to $V$. Define addition and scalar multiplcation in $V^n$ as follows: $(u_1,u_2,...,u_n)+(v_1,v_2,...,v_n)=(u_1+v_1, u_2+v_2,...,u_n+v_n)$...

hi, please tell me what do we mean when we say in quantum mechanics operators are linear and also vector space is also linear ?

Let S={x ∈ R; -π/2 < x < π/2 } and let V be the subset of R2 given by V=S^2={(x,y); -π/2 < x < π/2}, with vector addition ( (+) ). For each (for every) u ∈ V, For each (for every) v ∈ V with u=(x1 , y1) and v=(x2,y2) u+v = (arctan (tan(x1)+tan(x2)), arctan (tan(y1)+tan(y2)) )Note: The...

Hey, so my class just got into vector space and I don't have a clue what is going on. My teacher makes us go through the axiom for every problem but I can't visualize what is happening in our problems ( and our workbook makes me even more confused) It seems so... useless. The axioms. Why would...

Let B be a non-zero mx1 matrix, and let A be an mxn matrix. Show that the set of solutions to the system AX=B is not a vector space. I am thinking that I need to show that the solution is not consistent. In order to do so would I need to show that B is not in the column space of A?

I'm having trouble with a couple of things written in some notes I'm reading. Firstly, in stating examples of vector spaces, they say Trigonometric polynomials - Given n distinct (mod 2π) complex constants λ1,...,λn, the set of all linear combinations of eiλnz forms an n-dimensional complex...

Homework Statement Consider the set V = R^2 (two dimensions of real numbers) with the following operations of vector addition and scalar multiplication: (x,y) + (z,w) = (x+y-1, y+z) a(x,y) = (ax-a+1,ay) Show that V is a vector space Homework Equations None The Attempt at a Solution So...

Let \mathbb{Q}(\sqrt{2},\sqrt{3}) be the field generated by elements of the form a+b\sqrt{2}+c\sqrt{3}, where a,b,c\in\mathbb{Q}. Prove that \mathbb{Q}(\sqrt{2},\sqrt{3}) is a vector space of dimension 4 over \mathbb{Q}. Find a basis for \mathbb{Q}(\sqrt{2},\sqrt{3}). I suspect the basis is...

Question: Show that the set of all functions of the form f(x) = ax+b, with a and b real numbers forms a vector space, but that the same set of functions with a > 2 does not. Equations: the axioms for vector spaces Attempt: I think that the axiom about the zero vector is the one I need to use...

This isn't a homework problem, a classmate asked for a challenging proof to try and do and this was the one we were given. We started by trying to derive some rules from un-integratable functions but realized that that would take a long time and a lot of work. After some thinking we came up with...

Homework Statement In the real linear space C(-1, 1) with inner product (f, g) = integral(-1 to 1)[f(x)g(x)]dx, let f(x) = ex and find the linear polynomial g nearest to f. Homework EquationsThe Attempt at a Solution I understand that the best approximation for g is equal to the projection of...

Let ##V## be a real vector space and assume that ##V## (together with a topology and smooth structure) is also a smooth manifold of dimension ##n## with ##0 < n < \infty##, not necessarily diffeomorphic or even homeomorphic to ##\mathbb R^n##. Here's my question: Does this imply that addition...

Hi everyone! Can someone please tell me what is the proof of it? or Where can I find it? Thanks in advance. Best regards.

I'm confused by a set of problems my teacher created versus a set of problems in the textbook. My textbook states that "A vector space V over a field F consists of a set on which two operations (called addition and scalar multiplication, respectively) are defined so that for each pair of...

a metric is also used to raise/lower indices. g_{\nu \mu } x^{\mu} = x_{\nu} g^{ \nu \mu} x_{\mu} = x^{\mu} In general a metric [with lower indices] is a map from V_{(1)} \times V_{(2)} \rightarrow \mathbb{R} whereas the upper indices are the map from V^{*}_{(3)} \times V^{*}_{(4)}...

I'm confused about some of the notation in Hoffman & Kunze Linear Algebra. Let V be the set of all complex valued functions f on the real line such that (for all t in R) f(-t) = \overline{f(t)} where the bar denotes complex conjugation. Show that V with the operations (f+g)(t) = f(t) +...

Homework Statement Does the function: 4x-y=7 constitute a vector space? Homework Equations All axioms relating to vector spaces. The Attempt at a Solution x_n for example means x with the subscript n The book says that the function isn't closed under addition. So it continues...

Homework Statement Let b be a symetric bilinear form on V and A = \{ v\in V : b\left(v,v\right)=0\}. Prove that A is not a vector space, unless A = 0 or A = V. 2. The attempt at a solution If we suppose that A is a vector space then for every v,w\in A we must have...

I've been reading about algebraic geometry lately. I see that a lot of authors use ##V^\vee## to denote the dual space of a vector space ##V##. Is there any particular reason for this? The only reason I could think of is that this notation leaves us free to use ##R^*## to denote the units of...

A variant of a problem from Halmos : If AB=C and BA=D then explain why (C-D)^2 is commutative with all 2x2 matrices if A and B are 2x2 matrices. This result does not hold for any other nxn matrices where n > 2. Explain why. Edit: I tried to show that ((C-D)^2) E - E((C-D)^2) is identically...

Homework Statement For each of the following sets, either verify (as in Example 1) that it is a vector space, or show which requirements are not satisfied. If it is a vector space, find a basis and the dimension of the space. 6. Polynomials of degree ≤ 6 with a3 = 3. Homework Equations...