Vector space Definition and 540 Threads

Homework Statement Let \omega = \frac{1}{2} + \frac{\sqrt7}{2}i(a) Verify that \omega^2 = \omega - 2 (b) Prove that F = \{a + b \omega : a, b \in \mathbb{Q} \} is a field, using the usual operations of addition and multiplication for complex numbers. (c) Recall that we can think of F as a...

1 Suppose S is a set of n linearly independet in the n-dimensional vectorspace V. Prove that S is a basis for V. My try at this proof is: For S to be a basis for V it has to span V and the vectors in S needs to be linearly independent. But they have allreade sad that the vectors in S are...

Suppose V is a vector space over a field F that has multiplicative identity 1. Do we have to take, as an axiom, that 1\vec{v} = \vec v 1= \vec{v} for every \vec v\in V, or is this a direct consequence of other, more rudimentary vector space axioms?

I that this is probably another really simple question, but I would like some help learning how one starts a 'verification problem.' Verify that Fn is a vector space over F. I know that I just have to show that commutativity and scalar multiplication etc. are satisfied. But I am not used to...

Homework Statement Define V =R with vector addition a+b=ab and scalar multiplication za=a^z. Show that V is a vector space. Homework Equations a+b=ab, za=a^z The Attempt at a Solution I was able to check all the axioms but one, the additive inverse axiom where for all v in...

Homework Statement The set R^2 with addition defined by <x,y>+<a,b>=<x+a+1,y+b>, and scalar multiplication defined by r<x,y>= <rx+r-1,ry>. The answer in the back of the book says it is a vector space, but I am having trouble proving that 0+v=v and v+(-v)=0 Homework Equations The...

Homework Statement show that the collection of all ordered 3-tupples (x1,x2,x3) whose components satisfy 3x1 - x2 + 5x3 = 0 forms a vector space with the respect the usual operation of R3. Homework Equations 3x1 - x2 + 5x3 The Attempt at a Solution we tried it by addition and...

Homework Statement (i) Is the set of all mappings f:R3->R a)f(x,y,z)=ax+by+cz b)f(x,y,z)=ax+by+cz+d for a,b,c,d Є R, vector space, for the standard addition operation and scalar multiplication of function with real number? (ii) Is the set of all vectors in R3 which are collinear with the...

Definitions: A linear operator T on a finite-dimensional vector space V is called diagonalizable if there is an ordered basis B for V such that [T]_B is a diagonal matrix. A square matrix A is called diagonalizable if L_A is diagonalizable. We want to determine when a linear operator T on a...

Homework Statement use the subspace theorem to decide if the sets is a real vector space with respect to the usual operation the set of all solutions of the homogenous differential equation 7f''(x) +4f'(x) -6f(x) = 0 Homework Equations none The Attempt at a Solution try to...

how to proof if the solution set of a second order diffential equation af''+bf'+cf=0 is a real vector space w.r.t. the usual opeations?

can any1 pls tell me or explain the following..? 1.wat is the meaning of trivial solution? 2.wat is the difference between vector and vector space? 3.wat is vector space...? 4.why is the element in a field is called scalar? 5.how to illustrate a vector space over a field? 6.wat is...

Homework Statement Let V be a vector space over C of dimenson n . We view V also as a vector space over R by restricting the scalar multiplication of C on V to R .Show that dimR(V) = 2n Homework Equations The Attempt at a Solution I have to show that if x1,...xn form a basis of V...

Homework Statement Let V be a vector space over a field F and let x1,...xn \inV.Suppose that x1,...xn form a maximal linearly independent subset of V. Show that x1,...xn form a minimal spanning set of V. Homework Equations The Attempt at a Solution I knew that x1,...xn are linear...

Prove: the set of 3x3 symmetric matrices is a vector space and find its dimension. Well in class my prof has done this question, but I still don't quite get it.. Ok, first off, I need to prove that it's a vector space. The easy way is probably to prove that it contains the zero space and...

Apologies, have solved this question. Answer if useful for anyone: Basis= {e^(5x),e^(-10x)}. Homework Statement Consider the differential equation (2nd derivative of y wrt x) + 5(1st derivative of y wrt x) - 50y =0 Find a basis of the vector space of solutions of the above differential...

Hi, I'm a reading a thesis on Generalised Complex geometry and it mentions an object " det V" and "det V*", for a real vector space V, and its dual V*. Could anyone tell me what this notation means? I've been unable to find anything mentioning a determinant of an entire vector space, so I'm...

Homework Statement Let S be a set and let V be a vector space over the field F. LetV^S denote the set of all maps from S to V . We define an addition on V^S and a scalar multiplication of F on V^S as follows: (f+g)(s):=f(s)+g(s) and (af)(s):=a(f(s)) for any s belongs to S show that V^S is a...

Homework Statement Let S be a set and V a vector space over the field k. Show that the set of functions f: S --> k, under function addition and multiplication by a constant is a vector space. Homework Equations I think I need to show that (f+g)(x) = f(x) + g(x) and that (rf)(x) = rf(x)...

Homework Statement Let {e1,...,en} be a basis for a complex vector space X. Find a basis for X regarded as a real vector space. What is the dimension of X in either case? Homework Equations The Attempt at a Solution I'm really not sure where to begin with this question. Are the...

Homework Statement Let V be the real functions y=f(x) satisfying d^2(y)/(dx^2) + 9y=0. a. Prove that V is a 2-dimensional real vector space. b. In V define (y,z) = integral (from 0 to pi) yz dx. Find an orthonormal basis in V. The Attempt at a Solution part A: I integrated and got...

Homework Statement If W1 and W2 are subspaces of V, which is finite-dimensional, describe A(W1+W2) in terms of A(W1) and A(W2). Describe A(W1 intersect W2) in terms of A(W1) and A(W2). A(W) is the annihilator of W (W a subspace of vector space V). A(W)={f in dual space of V such that f(w)=0...

If V1 and V2 are both subspaces of a vector space V, then in order for their tensor product to be defined, does the intersection of V1 and V2 have to be 0?

[b]1. Homework Statement [/b Let's s denote the collection of all sequences in lR, let m denote the collection of all bounded sequences in lR, let c denote the collection of all convergent sequences in lR, and let Co denote the collection of all sequences...

Hi, good morning! I'm having trouble with vector space. Let there be α and β some given numbers. Prove that the set of all the real numbers f(x) so that: α*f(-2) + β*f(5) = 0 is a vector space ! Could someone please write a full solution for he axiom scalar multiplication?

Hi, I need help with this: Let V be a vector space (V may be infinite) and let W be a subspace of V, if "B" is a vector in V that doesn't belong to W, prove that if "A" is a vector in V such that "B" exists in the subspace WU{A} then "A" exists in the subspace WU{B}. I also have a...

Homework Statement Find a basis for V. Let W be a vector space of dimension 4. Let beta = {x1, x2, x3, x4 } be an ordered basis for W. Let V = {T in L(W) | T(x1) + T(x2) = T(x4) } Homework Equations L(W) is the set of linear transformations from W to W The Attempt at a Solution...

Homework Statement Let V be a vector space of dimension n. And the linear operators E=A^0, A^1, A^2, ... A^(n-1) are linearly independent. Prove that there exists a v in V such that V=<v, Av, A^2v, ..., A^(n-1)v> Homework Equations The Attempt at a Solution Here are something that...

Homework Statement Let V be a vector space over a field K. Let S be a set of vectors of V, S= {e_i : i in J} (i.e e_i in V for each i in J) where J is an index set. Prove that if S satisfies the following property, then S must be a basis. The property is: For every vector space W over...

I guess this is a bit of an interdisciplinary post... a friend of mine was looking at surds as a way of making a vector space with a basis as the prime numbers. Here's the construction: V = {Surds} = {qr|q is rational and positive, r is rational} The field that V is over is Q And we...

Homework Statement I have to show that l1, l2 and linfinity are norms The Attempt at a Solution Do you just go through the conditions for norm spaces ie: 1. ||x||>0, ||x|| = 0 iff x = 0 2.triangle inequality 3.||cx|| < |c|||x|| if the space satisfies these conditions it is a norm??

Please could someone help me with this question, thank you. Find dim[Ker(D^2 -D: P_3(F_3) ==>P_3(F_3))] Where dim is dimension, Ker is kernal D is the matrix 0100 0020 0003 0000 D^2 is the derivative of D is it equals 0020 0006 0000 0000 And F_3 is the field subscript3...

Hi, I have to find a vector space V with a real sub space U and a bijective linear map. Here my Ideas and my questions: If the linear map is bijective, than dim V = dim U Because U is a real sub space the only way to valid this constraint is if the dimension is infinity. I wrote...

Homework Statement Suppose V is a complex vector space and T \in L(V). Prove that there does not exist a direct sum decomposition of V into two proper subspaces invariant under T if and only if the minimal polynomial of T is of the form (z - \lambda)^{dim V} for some \lambda \in C. Homework...

Homework Statement Let V be a vector space, and suppose {v_1,...,v_n} (all vectors) form a basis for V. Let V* denote the set of all linear transformations from V to R. (I know from previous work that V* is a vector space). Define f_i as an element of V* by: f_i(a_1*v_1 + a_*v_2 + ...

Homework Statement Is the following a vector space? The set of all unit lower triangular 3 x 3 matrices [1 0 0] [a 1 0] [b c 1] Homework Equations Properties of vector spacesThe Attempt at a Solution I checked the properties of vector space (usual addition and scalar multiplication). I...

Homework Statement S is a subset of vector space V, If V is an 2x2 matrix and S={A|A is singular}, a)is S closed under addition? b) is S closed under scalar multiplication? Homework Equations S is a subspace of V if it is closed under addition and scalar multiplication...

Homework Statement Let V be the \mathbb{R}-vector space \mbox{Herm}_n( \mathbb{C} ). Find \dim_{\mathbb{R}} V. The Attempt at a Solution I'd say the dimension is 2n(n-1)+n=2n^2-n, because all entries not on the main diagonal are complex, so you have n(n-1) entries which you have to...

I'm trying to prove (as part of a larger proof) that the product of a m x n matrix M with column space R^m and a n x o matrix N with column space R^n, MN, has column space R^m. I'm not sure where to begin. What I'm thinking should be the right approach is to show that any solution to M augmented...

Homework Statement Find an expression for <P|X|P> in terms of P(x) defined as <x|P> (and possibly P*(x) ) Homework Equations X|P> = x|P> Identity operator: integral of |x><x| dx The Attempt at a Solution Ok...<P|X|P> add the identity = Integral [ <P|X|x> <x|P> dx ] = Integral [<P|x|x>...

Homework Statement Find a basis for V=\mathbb{C}^1, where the field is the real numbers. The Attempt at a Solution I'd say \vec{e}_1=(1,0), \vec{e}_2=(i,0) is a basis, because it seems to me that \vec{u}=a+bi \in V can be written as...

Hi everyone... I just wanted to know if you compute a cross product of two vectors in R^3, do you get a vector in R^3 or an actual value(say both vectors have actually values)... Another question. I did this in class but I wasn't sure how it would work. Let say I have a metrix 2x2. how do...

Recommend some good books to study vector space .

Homework Statement Suppose V1 (dim. n1) and V29dim. n2) are two vector subspaces such that any element in V1 is orthogonal to any element in V2.Show that the dimensionality of V1+V2 is n1+n2 Homework Equations The Attempt at a Solution The subspace V1 is spanned by n1 linearly...

Homework Statement Do functions that vanish at the end points x=0 and x= L form a vector space? What about periodic functions obeying f(0)=f(L)?How about functions that obey f(0)=4 Homework Equations The Attempt at a Solution We consider functions defined at 0<x<L.We define scalar...

Homework Statement "Consider all functions f(x) defined in an interval 0\leqx\leqL. We define scalar multiplication by a simply as af(x) and addition as pointwise addition: the sum of two functions f and g has the value f(x)+g(x) at the point x. The null function is simply zero everywhere...

Homework Statement Prove that V is infinite dimensional if and only if there is a sequence v_1, v_2,... of vectors in V such that (v_1,...,v_n) is linearly independent for every positive integer n. Homework Equations A vector space is finite dimensional if some list of vectors in it...

[/b]1. Homework Statement [/b] Let P_2 be the set of all real polynomials of degree no greater than 2. Show that both B:={1, t, t^2} and B':= {1, 1-t, 1-t-t^2} are bases for P_2. If we regard a polynomial p as defining a function R --> R, x |--> p(x), then p is differentiable, and D: P_2 -->...

The sum of two subspaces seems a simple enough concept to me, but I must be misunderstanding it since I don't understand why Axler gives an answer he does in Linear Algebra Done Right. Suppose U and W are subspaces of some vector space V. U = \{(x, 0, 0) \in \textbf{F}^3 : x \in...

One book defined a vector space as a set of objects that can undergo the laws of algebra "over" the field of scalars. But doesn't the laws of algebra also hold in a field? If so, wouldn't a field be a vector space also? Wouldn't that make the definition of a vector space meaningless as it uses...