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6.1 Vector Spaces and Subspaces Definition Let V be a set on which two operations, called addition and scalar multiplication, have been defined. V is called a vector space if the following axioms hold for all vectors u, v, and w in V and all scalars (real numbers) c and d. 1) u + v is in V 2) u + v = v + u 3) (u + v) + w = u + (v + w) 4) There exists an element 0 in V such that v + 0 = v 5) For each v in V, there is an element –v in V such that v + (-v) = 0. 6) cv is in V 7) c(u + v) = cu + cv 8) (c + d)v = cv + dv 9) c(dv) = (cd)v 10) 1v = v Examples 1) The set of m x n matrices with matrix addition and scalar multiplication is a vector space. 2) Z3 = {0, 1, 2} with special operations (integer modulo 3) is a vector space. 3) Let Σ be the set of all sequences, with the following addition and multiplication: If a a1 , a2 ,... then a b a1 b1 , a2 b2 ,... b b1 , b2 ,... ka {ka1 , ka2 ,...} Σ is a vector space. Examples 1) The set of all real-coefficient polynomials of degree three, together with usual addition and multiplication, is not a vector space. 2) The set of all rational numbers with standard addition and multiplication is a not vector space. However, the set of all real numbers (complex numbers), with standard addition and scalar multiplication, is a vector space 3) Z3 (the set of all vectors in three-dimensional space whose components are integers), with usual addition and multiplication is not a vector space. Theorem 1) 2) 3) 4) 0v=0 c0 = 0 (-1)v = -v If cv= 0 then c = 0 or v = 0 Examples and Notation The following sets, with standard addition and scalar multiplication, are vector spaces: 1) M = {M / M is a matrix} 2) P = {P / P is a polynomial} 3) F = {f(x) / f(x) is a function} 4) Mnxn = {M / M is an nxn square matrix} 5) Pn = {P / P is a polynomial of degree at most n} 6) D = {f(x) / f(x) is a differentiable functions} Definitions A subset W of a vector space V is called a subspace if W itself is a vector space with the same scalars, addition and scalar multiplications as V. Zero space Every vector space V has two subspaces: {0} and V. Let W be a subset of a vector space V. Then W is a subspace of V iff the following two conditions hold: 1) If u, v are in W, then u v is in W. 2) If u is in W and c is a scalar, then cu is in W. Examples Show that: 1) W = {A/ A is a 3x3 symmetric matrix} is a subspace of M . 2) W = { f(x) / f(x) is a solution to the equation 2y’ + 4y = 0} is a subspace of F . 3) W = {bx + cx2 / b,c are real numbers} is a subspace of P2 . 4) W = { 5 + bx + cx2 / b,c are real numbers} is NOT a subspace of P2 . 5) W = { f(x) / f(x) is a solution to the equation 2y’ + 4y – 7= 0} is NOT a subspace of F . 6) W = {an / an is a convergent sequence} is a subspace of Σ . 6.2 Linear Independence, Basis, and Dimension Terminology for a Vector Space V Linear combination: A vector v is called a linear combinatio n of vectors v1 , v 2 ,..., v n if there are scalars c1 , c2 ,..., cn such that v c1v1 c2 v 2 ... cn v n . For a set B v1, v2 ,..., vn Linear Dependence: There is one vector in B that can be written as a linear combination of the other vectors in B. Linear Independence: If c1v1 c2v2 ... cn vn 0 , then c1 c2 ... cn 0 Spanning set: any vectors in V can be written as linear combination of vectors in B. Basis: B is a linearly independent, spanning set for V. Dimension: dim V = number of vectors in a basis for V. A vector V is finite-dimensional if dim V is finite. Examples 1) Is this set { x, 2x - x2 , 3x + 2x2 } linear independent in P2 ? 2) Is this set { sin 2x, sin x, cos x } linear independent? 3) Show that: a) B = { x, 1 + x, x – x2 } is a basis for P2 . b) B’ = { 1, x, x2 ,…, xn } is a basis for Pn . c) Express 2 + 3x – x2 as a linear combination of vectors in B and B’ . Note: This is a unique representation with respect to each basis. Definition Let B v1 , v 2 ,..., v n be a basis for a vector space V. Let v be a vector in V, and write v c1v1 c2 v 2 ... cn v n . Then c1 , c2 ,..., cn are called coordinate s of v with respect to B, and the column vector v B c1 c 2 : cn is called the coordinate vector of v with respect to B. u vB uB vB cuB cuB Theorems Let V be a vector space with dim V = n. Then: 1. Any linearly independent set in V contains at most n vectors, and can be extended to a basis for V. 2. Any spanning set for V contains at least n vector, and can be reduced to a basis for V. 3. Any linearly independent set, or spanning set consisting of exactly n vectors is a basis for V. Examples 1) Is S = { x + 2, x – 2 } a basis for P2 ? 2) Extend S to a basis for P2 . 3) Find bases for the following vector spaces: a) W1 = { 1 + 2ax + 3bx2 – 4bx3 } is a basis for P2 . n b) W2 = { [1, 2a, 3b, -4b] } is a basis for R . c) W3 = 1 2a 3b 4b is a basis for M2x2 6.3 Change of Basis Examples 1 3 Let B v1 , v2 be a basis for R where v1 and v2 2 1 2 1 7 Let C u1 , u 2 be a basis for R where u1 and u 2 0 8 2 1) Find coordinate vector of x = [-3, 8] with respect to B. 2) Find coordinate vector of x = [-3, 8] with respect to C. 3) Find a matrix P such that P [x]B = [x]C Definition Let B u1 , u 2 ,..., u n and C v1 , v 2 ,..., v n be bases for a vector space V. The change - of - basis matrix from B to C is denoted by PCB and given by : PCB u1 C , u 2 C , ... , u n C