MATA33H3 Lecture Notes - Lecture 10: Idw Publishing, Royal Institute Of Technology, Triangular Matrix

(10 points) Let An M,(R) be given by (An)ij-?,j +6,HJ-5- det(An) F, where Fo F 1 and F- F-1F-2 are the Fibonacci numbers. 1.j. Prove that

Let V and W be finite dimensional vector s Let T V W and U W V be paces. linear and assume that dim(V) dim(W). rove that if U(T(r)) for all TE V then T (U(i)) y for all E W. Hint 1: Prove that nu 0 and use the given assumptions to conclude that T is onto. Hint 2: Since T is onto any i E W can be written as ij F T(2) for some 2 E V.

(14 points) The vector space Mmxn(R) has dimension mn. Let Ei denote the matrix whose (i,j)th entry is 1 and all other entries are 0. The standard basis E(m, n) of Mmxn consists of the mn matrices {Eij: 1