MATH 219 Final, Summer 2004

of RELROT Mathematics 219 Final Exam. (August 16, 2004) Time: 130 minutes 1. Let A and B be two matrices with 7 columns such that rank (A)=2, rank (B)=3, rank gh B Let A*, B*, C* be the null spaces of A, B and 4) respectively. (i) Use the rank-nullity theorem to find the B dimensions of A*, B*, and C* {ii) Deduce that A* +B* = R’. 2. If {a, b, c} is a basis of a vector space V, show that {atc, b+c, c} is also a basis of V. 3. Let f£:V > W be linear transformation of vector spaces such that { f(v,), , fv, )} are linearly independent. Show that {y,, ,v,, }are linearly independent vectors in V. 4. Prove the following theorem Without using the rank-nullity Theorem. Theorem: Let f:/ > W be linear transformation of vector spaces. If f is 1-1 and dimV=dimW, then f is an isomorphism. (Reminder: You may use any result proven in class unlessyou are asked to prove it) 5. Finda 5x5 matrix A such that rank A=3 and rank A? = 2. (Hint: Apply the Linear Extension Theorem) 6. Show that similar matrices have the same determinant, the same trace, and the same eigen values. 7. If 4 is aneigen value of T°, show that 2 or ~2 is an eigenvalue of 7. (Hint: Use upper triangulization) 3°10 8. Let A=|6 2 0 (i) Find the eigen values of A and a basis for each eigen space of A. 00 5 (ii) Show that A is diagonalizable and find the exact relation between A,P and D. (Do not calculate PT). 9. (a) What do we know about arbitrary symmetric matrices regarding eigenvalues & diagonalization ? (b What do we know about the minimal polynomial with respect to diagonalization? (c) Apply the Cauchy-Schwarz inequality the inner product space of continous functions on [a, b] (with the well- 1 known way of “dotting” functions), then find an upper bound on f xe dy , 0 10. True-False._ If false, give a counter example (a) Ifaset of 3 vectors (say in R*) have the property that each 2 of them are linearly independent, then this set is linearly independent. (b) For an mxn matrix 4, A & RRE(A) have the same null space & row space. (c) Foran mxn matrix A, A & RRE(A) have the same column space. (d) A 4x4 matrix with 4 distinct eigenvalues is diagonlizable. 11. Suppose every vector in V is an eigen vector for T where T is a linear operator on a vector space V, show that T is a scaler multiple of the identity I. (Hint: Use a, b, a+b) gy 12. Let V=A+B=A+C where A and B are subspaces of a vector space V. Suppose dimA=, dimB=5, and ANB =0=ANC. Show that dimC=S.