Vector space Definition and 530 Threads

decide whether this is a vector space or not a(x,y,z) = (2ax,2ay,2az) all the addition axoims hold easily for the scalar multiplications axioms for some real scaral a a(x,y,z) = (ax,ay,az) \in 2(ax,ay,az) a(x_{1}+x_{2},y_{1}+y_{2},z_{1}+z_{2}) = a(x_{1},y_{1},a(z_{1}) +...

Hi I'm given the following assignment which deals with to looks like an L-normed vectorspace: Prove that, |f(y_1) - f(y_2)| \leq || y_1 - y_2|| To prove this do I approach the above as a triangle inequality or as a cauchy-swartz inequality? Best Regards, Fred

We have x=x1(1,0,0) + x2(0,1,0) + x3(0,0,1) to represent R^3. That's a finite dimensional vector space. So what do we need infinite dimensional vector space for? Why do we need (1,0,0,...), (0,1,0,0,...), etc. bases vectors to represent R^1 ?

Two m x n matrices A and B are called EQUIVALENT (writen A ~e B if there exist invertible matracies U and V (sizes m x m and n x n) such that A = UBV a) prove the following properties of equivalnce i) A ~e A for all m x n matracies A ii) If A ~e B, then B ~e A iii) A ~e B and B~e C, then...

Okay, so i have this problem in my text, and I've almost figured it out (i think) but i need a little help "Let V be the set of all polynomials of degree 3. Define addition and scalar multiplication pointwise. Prove that V with respect to these operations of addiont and scalar multiplication...

Hi can someone assist me with the following question? Q. Let V be a finite dimensional real vector space with inner product < , > and let W be a subspace of V. Then the orthogonal complement of W is defined as follows. W^o = \{ v \in V: < v,w > = 0,w \in W\} Prove the following...

Hello, I am having trouble with particular algebra question. I don't know where to start and it would be greatly appreciated if someone could point me in the right direction. Here is the questoin: Let V be a vector space, where T is a linear map of V prove if T^2 = 0 then I - T is...

Hi everybody, I have one question about vectors of R^3: First of all, a point is described by its co-ordinates (x,y,z). A vector r is described in this way: r=ax+by+cz ,where {x,y,z} is the standard basis (the numbers a,b,c are the "coordinates" of the vector). But i have seen in several...

Okay, so I am doing this homework question, and its bothering me, so i thought perphaps somebody can help me out. " Let P4 denote the vector space of all polynomials with degree less than or equal to 4 and real coefficients. Describe percisely as you can the linear span of set {x^2 – x^4...

I understand that the definition of the number of dimensions of a vector space, but somehow that doesn't really help me with physical dimensions. How in practice do we know that our space is 3-dimensional?

Hello... I've been doing some home work on Vector Spaces and Vector Subspaces and I need help solving a problem... Can somebody please help me? Consider the differential equation f'' + 5f' + 6f' = 0 Show that the set of all solutions of this equation is a vector subspace of the...

Second week in Linear Algebra... My homework involves of identifying all failing Vector Space Axioms for various sets of vector spaces. I did fine with a "regular" set like (x,y,z) which has an operation like k(x,y,z)=(kx,y,z). I have worked through all 10 of the axioms, comparing left sides...

Let V = {(a1, a2, ..., an): ai in C for i = 1, 2, ... n}; (C=complex numbers) ; so, V is a vector space over C. Is V a vector space over the field of real numbers with the operaions of coordinatewise addition and multiplication? I thought the answer to this question is No since after we...

While reading text, I had a question which I can not resolve by myself. Please Help me! it reads, The empty set( a vector space with no elements) is denoted as & (This symbol doesn't matter for the sake of argument, I don't know how to write the Zero with a line in the middle). I can...

Hi what are the differences between affine and vector spaces ? Please can you give me examples. thanks roger

Is the following a vector space: the set of all ordered triples of real numbers, {x1,x2,x3)}, usual addition, and r(x1,x2,x3)=(0,0,0), all numbers r I think this is a vector space since it is the vector (0,0,0), but I'm not sure how to show the work for it. Thanks in adv.

Let V be a 5 dimensional vector space, and let \Delta be a determinant formon V. Given \Delta(b1, b2, b3, b4, b5)= -3 How do I find \Delta(b4, b3, b5, b1, b2)?

In the vector space P_4 of all polynomials of degree less than or equal to 4 we define the first five Tchebychev polynomial as p_0(x) = 1 p_1(x) = x p_2(x) = 2x^2 - 1 p_3(x) = 4x^3 - 3x p_4(x) = 8x^4 - 8x^2 + 1 To show that B={p_0, p_1, p_2, p_3, p_4} is a basis of P_4, do I put them...

This is killing me that I can't see this. Why is it two norms on a finite dimensional Vector space X are equivilant if and only if there exist positive real constants c_1,\, c_2 such that \forall x\in X, \|x\|_2 \le c_1 \|x\|_1 and \|x\|_1 \le c_2 \|x\|_2. Here equivilant means that a...

I would really appreciate if anyone could help me with this problems. V is a vector space on R and v1, v2, v3 e V are linearly independant. If w1 = v1 + kv2, w2= v2 - 2kv3 and w3= v3 - 4kv, find k so w1, w2, w3 are linearly dependant. I tried it and got k=0 and I think it's wrong :mad:

So if we let x>0, For which 0<p<=infinity is {1/n^x} an element of l^p? If x=1, then 1/n^x is clearly an element of l^p for p>=2, since for all these vector spaces, the series of 1/n will converge? But if x<1, then in it seems that only for p=infinity, will {1/n^x} be an element of l^p. Is...

I had this question in my book asking me to show these things in detail, but it seems easy yet i don't understand why teacher said it was a little difficult: 1) Prove that R^2(with the rules of addition and scalar multiplication) is a vector space and find (zero vector)? 2)Deduce from the...

Hi everybody, In vector spaces we define two operations, addition and scalar multiplication. Scalar multiplication is distributive over addition. This can define the order of operations in the vector space? I mean when we have an expression to calculate, we know that we firstly calculate...

Hi everyone, general question: is a solution set for a particular system a vector space? I know it can be if there is a unique solution, but is it generally true? Could someone explain, please? Thanks.

If I have a finite dimensional inner product space V = M_{n \times n}(\mathbb{R}), then one basis of V is the set of n² (n x n)-matrices, \beta = \{E_1, \dots , E_{n^2}\} where E_i has a 1 in the i^{th} position, and zeroes elsewhere (and by i^{th} position, I mean that the first position is the...

I don't totally understand spanning sets... Can anyone explain this problem to me: let V = set of all polynomials with degree of 2 or less (a vector space_ let S = {t + 1, t^2 + 1, t^2 - t} Does S span V? I know that (t^2 + 1) - (t + 1) = t^2 - t But I just don't see what that...

Ok another question: 13. The set of solutions to the system of linear equations a - 2b + c = 0 2a - 3b + c = 0 is a subspace of R^3. Find a basis for this subspace The book claims one of the possible bases is (1, 1, 1) but I don't see how. I mean I realize a = b = c from the...

vector space... help! i just got into vector spaces and i am really stump. okay from teh definition of vector space, it says something... "w/ the operation of mult by a number and addition. more briefly, we refer to V as a real vector space." so from a question from an exercise: determine...

how do you prove that if v is an element of V (a vector space), and if r is a scalar and if rv = 0, then either r = 0 or v = 0... it seems obvious, but i have no idea how to prove it...

The 3 dimensional space that we inhabit must have a basis of 3 vectors which is fair enough. But in my partial differential equations class in which Fourier series was introduced, it was said that piecewise smooth function space has a basis of an infinite number of vectors. If there is a...