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6.1 Vector Spaces and Subspaces

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 i n V and all scalars (real numbers) c and d .

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6.1 Vector Spaces and Subspaces

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  1. 6.1 Vector Spaces and Subspaces

  2. Definition Let V be a set on which two operations, called addition and scalar multiplication, have been defined. Vis called a vector space if the following axioms hold for all vectorsu, v,and win V and all scalars (real numbers) c and d. u + v is in V u + v = v + u (u + v) + w = u + (v + w) There exists an element 0 in V such that v + 0 = v For each v in V, there is an element –v in V such that v + (-v) = 0. cvis in V c(u + v) = cu + cv (c + d)v = cv + dv c(dv) =(cd)v 1v= v

  3. 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 Σbe the set of all sequences, with the following addition and multiplication: If then  Σ is a vector space.

  4. 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.

  5. Theorem 0 v = 0 c0= 0 (-1)v = -v Ifcv= 0 then c = 0 or v = 0

  6. 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}

  7. 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:

  8. 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 .

  9. 6.2 Linear Independence, Basis, and Dimension

  10. Terminology for a Vector Space V • Linear combination: • Linear Dependence: There is one vector in B that can be written as a linear combination of the other vectors in B. • Linear Independence: • 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.

  11. Examples 1) Is this set { x, 2x-x2 ,3x+ 2x2 } linear independent in P2? 2) Is this set { sin 2x, sinx,cosx } 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. 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.

  12. Definition

  13. Theorems Let V be a vector space with dim V = n. Then: Any linearly independent set in V contains at most n vectors, and can be extended to a basis for V. Any spanning set for V contains at least n vector, and can be reduced to a basis for V. Any linearly independent set, or spanning set consisting of exactly n vectors is a basis for V.

  14. 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. b) W2 = { [1, 2a,3b, -4b]} is a basis for Rn. c) W3 = is a basis for M2x2

  15. 6.3 Change of Basis

  16. Examples 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

  17. Definition

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