Highlights Reminder. The first examination is Friday, June 9. Math 304 Linear Algebra From last time: I A set of vectors is linearly independent if none of the vectors is in the span of the others. Harold P. Boas I A set of vectors is linearly dependent if some non-trivial [email protected] linear combination of the vectors equals the 0 vector. Today: June 7, 2006 I the notions of basis and dimension I changing coordinates Basis Example: exercise 10 on page 151 3 A basis for a vector space is: Extract a basis for R from the following vectors: I a linearly independent set of vectors that spans the vector 1 2 1 2 1 space x1 = 2 , x2 = 5 , x3 = 3 , x4 = 7 , x5 = 1 I equivalently, a maximal linearly independent set of vectors 2 4 2 4 0 I equivalently, a minimal spanning set for the vector space Solution. The vectors x1 and x2 are linearly independent, but 3 Example. The standard basis for R consists of the three they do not form a basis for R3 because they do not span R3. 1 0 0 The vectors x1, x2, and x3 do not form a basis because they vectors e = 0 , e = 1 , and e = 0 . 1 2 3 are linearly dependent (x3 = x2 − x1). Vectors x1, x2, and x4 do 0 0 1 not form a basis because they are linearly dependent: One nonstandard basis for R3 consists of the three vectors 1 2 2 1 2 1 1 4 7 det 2 5 7 = 0. However, 2 5 1 = −2 6= 0, so v1 = 2, v2 = 5, and v3 = 8. 2 4 4 2 4 0 3 6 0 3 vectors x1, x2, and x5 do form a basis for R . Dimension Change of basis: an example 1 −1 Let u = and u = be a nonstandard basis for R2, 1 1 2 1 The dimension of a vector space is the number of vectors in a 5 and let e and e be the standard basis vectors. Let x = . basis. (All bases of a vector space have the same number of 1 2 3 vectors.) Then x = 5e1 + 3e2. In other words, 5 and 3 are the Examples. coordinates of the vector x with respect to the standard basis. 3 I The dimension of R equals 3. Since x = 4u1 − u2, the coordinates of x with respect to the 2×5 u basis are 4 and −1. I The dimension of the space R of 2 × 5 matrices equals 10. Question. How are the u coordinates and the standard coordinates related? I The dimension of the space P3 of polynomials of degree less than 3 equals 3. (One basis is 1, x, x2.) 5 1 −1 4 = x = 4u − u = 1 2 − I The function space C[0, 1] is infinite dimensional. (Any 3 e 1 1 1 u finite subset of the functions 1, x, x2, x3, . is linearly independent.) The transition matrix between u coordinates and standard coordinates is the matrix whose columns are the standard coordinates of the u basis vectors. The inverse matrix gives the change from standard coordinates to u coordinates. Example continued 5 4 Consider another nonstandard basis v = , v = . 1 6 2 5 How are coordinates with respect to the u basis related to coordinates with respect to the v basis? 5 4 The matrix whose columns are v and v transforms 6 5 1 2 v coordinates to standard coordinates, and the inverse matrix 5 −4 transforms from standard coordinates to −6 5 1 −1 v coordinates. Since transforms from u coordinates 1 1 to standard coordinates, the product matrix 5 −4 1 −1 1 −9 = −6 5 1 1 −1 11 transforms from u coordinates to v coordinates. 1 −9 4 13 So the v coordinates of x are = . − − − 1 11 1 u 15 v.