MATH2111 Tutorial 9 Linear Independence As in Rn , a set of vectors {v1 , v2 , . . . , vk } in a vector space V is linearly independent if t1 v1 + t2 v2 + · · · + tk vk = 0 implies t1 = t2 = · · · = tk = 0. A set of vectors that is not linearly independent is said to be linearly dependent. 1. Let f and g be continuous functions on [a, b], and assume that f (a) = 1 = g(b) and f (b) = 0 = g(a). Show that {f, g} is linearly independent in C[a, b]. 2. Suppose A is an n× n matrix such that Ak−1 6= O but Ak = O for some positive integer k. Show that I, A, A2 , . . . , Ak−1 is linearly independent in Mn×n . Prepared by Leung Ho Ming Homepage: http://ihome.ust.hk/~malhm 1 Basis for a Vector Space A set B of vectors in V is a basis for V if B is linearly independent and span B = V . The dimension of V , denoted by dim V , is the number of vectors in a basis for V . 3. Suppose S = {A1 , A2 , · · · , Amn } is a basis for Mm×n . If U is an invertible m × m matrix, V is an invertible n × n matrix, show that T = {U A1 V, U A2 V, . . . , U Amn V } is a basis for Mm×n . 4. The set U = A:A 1 −1 1 1 = 0 −1 1 A is a subspace of M2×2 . Find a basis for U and the dimension of U . 0 2 Procedures for Finding Basis for Null Spaces, Column Spaces and Row Spaces Suppose A is an m × n matrix. After solving Ax = 0, a set of basic solutions is a basis for Nul A. If A is carried to a row-echelon form R, the nonzero rows of R form a basis for Row A. The columns of A corresponding to the pivot positions of R form a basis for Col A. Note that dim(Col A) = dim(Row A) = rank A, and we have the Rank Nullity Theorem saying rank A + dim(Nul A) = n. 1 −1/2 0 2 −1 1 1 0 5/2 −3/2 0 −2 1 1 0 1 and rref AT = 0 1 1/2 3/2 5. Let A = 4 −2 3. rref A = 0 0 0 0 0 0 0 0 0 0 −6 3 0 (a) Find a basis for Nul A. (b) Find two bases for Col A. (c) Find two bases for Row A. 3 Coordinates Relative to a Basis Let B = {b1 , . . . , bk } be an ordered basis for V (where the order in listing is taken into account). Every v ∈ V has a T unique representation v = c1 b1 +· · · ck bk , the coordinates of v relative to B is the column vector [v]B = [c1 c2 · · · ck ] k in R . The mapping T : V → Rk defined by T (v) = [v]B is a one-to-one and onto linear transformation, whose inverse is also linear. 1 1 1 0 0 0 1 0 1 2 , , is a basis for M2×2 . Find [v]B if v = . 6. B = , 0 0 1 0 1 1 0 1 −1 0 T 7. B = 1 + t, 1 − t, t − t2 is a basis for P2 . Find p(t) if [p(t)]B = [1 2 3] . 2 1 1 8. Let B = 3 , 2 , 3 be a basis for R3 , then the B-coordinate mapping is a linear transformation from 2 1 2 R3 to R3 . Find the standard matrix of the B-coordinate mapping. 4

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