Vector space Definition and 540 Threads

If S\subseteq V and V is a vector space, then S is a vector space. Assume S isn't a vector space. Since S isn't a vector space, then V isn't a vector space; however, V is a vector space. By contradiction, S is a subspace. Correct?

In other words, why is dim[{0}]=0. My math professor explained that since the 0 vector is just a POINT in R2 that the zero subspace doesn't have a basis and therefore has dimension zero. This is not satisfactory. For example, I know R2 has a dimension 2, P_n has dimension n+1, M_(2,2) has...

Homework Statement Show that the closed unit ball {x E V:||x||≤1} of a normed vector space, (V,||.||), is convex, meaning that if ||x||≤1 and ||y||≤1, then every point on the line segment between x and y has norm at most 1. (hint: describe the line segment algebraically in terms of x and y...

Homework Statement My notes has the following statement, but I seem to have forgotten to write down the conclusion of the statement before my professor erased it from the board. "Any vector space V there will be a basis except for 1 type of space: " Any ideas as to what that 1 type of...

im trying to get a homeomorphism between a 1-dim vector space and R, but independent of the basis. Any ideas?

Homework Statement Suppose V is a vector space with operations + and * (under the usual operations) and W = {w1, w2, ... , wn} is a subset of V with n vectors. Show Span{W} is a subspace of V. The attempt at a solution I know that to show a set is a subspace, we need to show...

Homework Statement Sorry for the vague title! Let R denote the set of real numbers, and F(S,R) denote the set of all functions from a set S to R. Part 1: Let \phi be any mapping from a set A to a set B. Show that composition by \phi is a linear mapping from F(B,R) to F(A,R). That is...

I think, in case it is wrong, I proved the the first vector space axiom for 3 x 3 magic squares; however, there has to be an easier way to do what I did. This pdf has been removed. Go to page 2 of the discussion for an updated version. I attached a pdf file due to I can create the...

Since I can't copy and paste from maple into this message w/out losing formatting, I attached a pdf with all the work. I am having trouble proving axiom 1 of two general magic square matrices added together; plus, I am not sure if my set notation is entirely correct.

Homework Statement Let V be a finite dimensional normed vector space and let U= L(V)*, the set of invertible elements in L(V). Show, f:U-->U defined by f(T)= T-1 is differentiable at each T in U and moreover, Df(T)H = -T-1HT-1 where Df(T)= f'(T). Homework Equations Apparently...

Find the basis of the vector space (1,2)^T; (-1,1)^T When I solve the matrix, I obtain x1=0 and x2=0 x=(0,0)^T. Can a basis be two 0 column vectors? Thanks for the help.

Homework Statement let T(V)=V be a linear map, where V is a finite-dimensional vector space. Then T^2 is defined to be the composite TT of T with itself, and similarly T^(i+1) = TT^i for all i >=1. Suppose Rank (T) = Rank (T^2) Homework Equations a) prove that Im(T) = Im(T^2) b) for...

Show that the functions (c_{1}+c_{2}sin^{2}x+c_{3}cos^2{x}) form a vector space. Find a basis of it. What is its dimension? My answer is that it's a vector space because: (c_{1}+c_{2}sin^{2}x+c_{3}cos^2{x})+(c'_{1}+c'_{2}sin^{2}x+c'_{3}cos^2{x})...

Homework Statement Let U and V be vector spaces of dimensions of n and m over K and let Hom(subscriptK)(U,V) be the vector space over K of all linear maps from U to V. Find the dimension and describe a basis of Hom(subscriptK)(U,V). (You may find it helpful to use the correspondence with mxn...

I am not sure if my #4 holds and I don't know how to approach #7. My Axioms are below the general axioms. {∀ x ϵ ℝ+ : x>0} Define the operation of scalar multiplication, denoted ∘, by α∘x = x^α, x ϵ ℝ+ and α ϵ ℝ. Define the operation of addition, denoted ⊕, by x ⊕ y = x·y, x, y ϵ ℝ+. Thus...

Homework Statement Calculate ||1,1,1||in R3 Calculate ||1,1,1,1|| in R4. Calculate ||1,1,...,1|| in Rn. Homework Equations All I have in this problem is that, Where do I start? The Attempt at a Solution

{∀ x ϵ ℝ+ : x>0} Define the operation of scalar multiplication, denoted ∘, by α∘x = x^α, x ϵ ℝ+ and α ϵ ℝ. Define the operation of addition, denoted ⊕, by x ⊕ y = x·y, x, y ϵ ℝ+. Thus, for this system, the scalar product of -3 times 1/2 is given by: -3∘1/2 = (1/2)^-3 = 8 and the sum of 2 and...

{∀ x ϵ ℝ+ : x>0} Define the operation of scalar multiplication, denoted ∘, by α∘x = x^α, x ϵ ℝ+ and α ϵ ℝ. Define the operation of addition, denoted ⊕, by x ⊕ y = x·y, x, y ϵ ℝ+. Thus, for this system, the scalar product of -3 times 1/2 is given by: -3∘(1/2)= (1/2)^-3 = 8 and the sum of 2 and 5...

Homework Statement I need to prove that the hermitian matrix is a vector space over R Homework Equations The Attempt at a Solution I know the following: If a hermitian matrix has aij = conjugate(aji) then its easy to prove that the sum of two hermitian matrices A,B give a hermitian...

Hi. please anyone help me with vector spaces and the way to prove the axioms. like proving that (-1)u=-u in a vector space.

Homework Statement find the rank and nullity of the linear transformation T:U -> V and find the basis of the kernel and the image of T Homework Equations U=R[x]<=5 V=R[x]<=5 (polynomials of degree at most 5 over R), T(f)=f'''' (4th derivative) The Attempt at a Solution Rank = 2...

Homework Statement erm, I just want to know, what is the basis for a zero vector space? Homework Equations The Attempt at a Solution is it the zero vector itself? but if that's the case, then the constant alpha could be anything other than zero, which means the zero vector is not...

Homework Statement V=R^{4}\ and\ a^{\rightarrow}, b^{\rightarrow}, c^{\rightarrow}, d^{\rightarrow}, e^{\rightarrow} \in V. (I'll drop the vector signs for easier typing...) a = (2,0,3,0), b = (2,1,0,0), c = (-2,0,3,0), d = (1,1,-2,-2), e = (3,1,-5,-2) Let\ U \subseteq V be\...

Homework Statement Find the dimnesion and a basis of vector space V Homework Equations V is the set of all vectors (a,b,c) in R^3 with a+2b-4c=0 The Attempt at a Solution (4c-2b,b,c) = b(-2,1,0) + c(4,0,1) so {(-2,1,0),(4,0,1)} is the basis of the SUBSPACE of V right? how do I...

Determine whether the given set S is a subspace of the vector space V. A. V=P5, and S is the subset of P5 consisting of those polynomials satisfying p(1)>p(0). B. V is the vector space of all real-valued functions defined on the interval [a,b], and S is the subset of V consisting of those...

Homework Statement Prove that the additive inverse -v of an element v in a vector space is unique. Homework Equations Additive Inverse in V For each v in V, there is an element -v in V such that v + (-v) = 0. The Attempt at a Solution Assume that the additive inverse is not...

Homework Statement Prove that the additive identity in a vector space is unique Homework Equations Additive identity There is an element 0 in V such that v + 0 = v for all v in V The Attempt at a Solution Assume that the additive identity is NOT unique, then there exists y...

Hi, given a complex vector space with a hermitian inner product, how is the cosine of the angle between two vectors defined? I tried to follow a similar reasoning as in the real case and I got the following: cos(\theta)=\mathcal{R}e \frac{ \left\langle u,v\right\rangle}{\left\|u\right\|...

dear all,we know that active transformation refers to action of changing vectors keeping the operators unchanged whereas passive transformation refers to change of operator components keeping vectors unchanged. what i cannot understand(i am just starting quantum mechanics)is in the former if we...

He gets only the positive vectors. But I don't get which is not a vector space. What I understand is vector space maybe a R^2, R^3 or R^n. Can anyone here explain it more clearly? I don't get what he said. http://www.youtube.com/watch?v=JibVXBElKL0" @ 29:55

In short: does every vector space have a "standard" basis in the sense as it is usually defined i.e. the set {(0,1),(1,0)} for R2? And another example is the standard basis for P3 which is the set {1,t,t2}. But for more abstract or odd vector spaces such as the space of linear transformations...

Homework Statement a b c 0 b 8 0 0 c Homework Equations 10 axioms to determine vector space: 1. If u and v are objects in V, then u + v is in V. 2. u + v = v + u 3. u + (v + w) = (u + v) + w 4. There is an object 0 in V, called a zero vector for V, such that 0...

Homework Statement Prove that V is cyclic relative to a linear transformation T, T:V->V if and only if the minimal polynomial of T is the same as the characteristic polynomial of T. Homework Equations The Attempt at a Solution i have finished the => direction (proved that if...

Homework Statement Let T be the set of all ordered triples of real numbers (x,y,z) such that xyz=0 with the usual operations of addition and scalar multiplication for R^3, namely, vector addition:(x,y,z)+(x',y',z')=(x+x',y+y',z+z') scalar multiplication: k(x,y,z)=(kx,ky,kz) Determine...

Homework Statement The set of all nonsingular 3x3 matrices does not form a vector space over the real numbers under addition. Why? Homework Equations A vector space over F, under addition, is a nonempty set V such that A1 Addition is associative A2 Existence of 0 A3 Existence of negative A4...

I have a quick question about vector spaces. Consider the vector space of all polynomials of degree < 1. If the leading coefficient (the number that multiplies x^{N-1}) is 1, does the set still constitute a vector space? I am thinking that it doesn't because the coefficient multiplying...

Hi, I am currently working through 'Schutz-First course in General Relativity' problem sets. Question 2 of Chapter 3, asks me to prove the set of one forms is a vector space. Earlier in the chapter, he defines: \tilde{s}=\tilde{p}+\tilde{q} \tilde{r}=\alpha \tilde{p} To be...

Homework Statement Does this set describe a vector space? Te set of all solutions (x,y) of the equation 2x + 3y = 0 with addition and multiplication by scalars defined as in R^2.Homework EquationsAssociativity of addition u + (v + w) = (u + v) + w. Commutativity of addition v + w = w + v...

Hello, I'm studying linear algebra and wanted to know what is the difference between a "vector space" and a "vector space over field F". I know that a vector space over field F satisfies the 8 axioms, but does a vector space satisfy this also?

Homework Statement In the 4-D Minkowski vector space [you can think of this as the locations of events in space-time given by (t, x, y, z)] consider the vectors pointing to the following events: (4ns, -1m, 2, 7) and (2ns, 3m, 1m, 9m) (a) Find the distance between the events. (b) Find the...

Homework Statement Is the set of all complex solutions to the differential equation \frac{d^2 y}{d x^2} + 2\frac{d y}{d x} - 3 y = 0 If so, find a basis, the dimension, and give the zero vector Homework Equations The Attempt at a Solution I solved the equation and got the...

vector space proof?? Let V = ((a1,a2): a1,a2 \in R). For (a1,a2), (b1,b2) \in V and c \in R, define (a1,a2) + (b1,b2) = (a1 + 2b1, a2 + 3b2) and c(a1,a2) = (ca1, ca2). Is V a vector space over R with these operations? Justify your answer. Does this set hold for all the eigth...

Homework Statement Determine whether or not the given set is a subspace of the indicated vector space: Functions f such that [integral from a to b]f(x)dx = 0; C[a,b] (not sure how to do the coding for integrals) Homework Equations to be a subspace it must follow these axioms: (i) if x and y...

Homework Statement Show that if V is a vector space, a is any scalar and u is a member of V then 1) (-1)x = -x 2) a(-u) = -au 3) -(-u) = u Homework Equations The ten axioms of vector space. The Attempt at a Solution I have solved a0 = 0, but I couldn't figure out how to start answering these...

Just wondering. Suppose we some plane, any plane like S = \{ (x_1, x_2, x_3) \in F^{3} \ : \ x_1 + 5x_2 + 3x_3 = 0 \} where F is either \mathbb{R} or \mathbb{C} . We know that S is a vector space (passes the origin). We know that (0,0,0) is the additive identity and it should be unique by...

1. The problem statement Let W = {(x, y, z, t): x + y + 2z - t = 0} be a vector space under R^4. Find a basis of W over R. 2. The attempt at a solution To me I would think that the vector space itself could its own basis, but I know I'm probably way off. I also tried solving x = t - y...

I'm considering the problem: Given c \in \bold{F}, v \in V where F is a field and V a vector space, show that cv = 0, v \neq 0 \ \Rightarrow \ c = 0 I've been wrapping my head around this one for a while now but I can't seem to get it. Proving that if cv = 0 and v \neq 0 implies v = 0 is...

Homework Statement Claim: The solution space of a linear homogeneous PDE Lu=0 (where L is a linear operator) forms a "vector space". Proof: Assume Lu=0 and Lv=0 (i.e. have two solutions) (i) By linearity, L(u+v)=Lu+Lv=0 (ii) By linearity, L(au)=a(Lu)=(a)(0)=0 => any linear...

Let X be a normed vector space. If C is a closed subspace x is a point in X not in C, show that the set C+Fx is closed. (F is the underlying field of the vector space).

Homework Statement Prove: If V is an n-dimensional vector space of a finite field, and if 0 <= m <= n, then the number of m-dimensional subspaces of V is the same as the number of (n-m)-dimensional subspaces. The Attempt at a Solution Well here's a sketch of my argument. Let U be an...