Question

s={(8.60) :) :) is a basis of M3x2(R)? (d) (1 point) The set = {(1 9:(. :) : 6 1) (1 1) (1 :) :()} is linearly independent. (e) (1 point) For a linear transformation A:R" + Rd the dimension of the nullspace is larger than d. (f) (1 points) Let AC M4x4 be a diagonal matrix. A is similar to a matrix A which has eigenvalues 1,2,3 with algebraic multiplicities 1,2, 1 and geometric multiplicities 1,1, 1 respectively. 8. Consider the ODE y+y=0. (a) (3 points) Show that the solutions to this ODE form a subspace of the vector space C(R). (b) (1 point) What is the smallest that the dimension of this subspace could be? Explain (Hint: e-- is a solution to the ODE). (c) (1 point, HARDER) What is the largest that the dimension of this subspace could be? Explain. 9. We saw how JNF generalizes the notion of diagonalization, and we will now look at a similar concept which generalizes the notion of an inverse. The matrix pseudo-inverse of A € Minxm(R) is the matrix A+ e Mmxm(R) which satisfies the following four properties (a) AA+A= A (b) A+AA+ = A+ (c) (AA+)? = AA+ (d) (A+A)" = A+A Quickly convince yourself that this is indeed a generalization of the notion of A-? The following is true: Any matrix A € Mmxm(R) has a unique pesudo-inverse A+ e MmX(R). (9) (note that this matrix does not have (a) (3 points) Find the unique pseudo-inverse of A = an inverse). (b) (1 point) Prove that (AB)+ = B+A+. (c) (1 point) Prove that (A+)+ = A. (d) (1 point) Prove that A+ = A-1 when A is invertible. 10. Given A= (+ 2) (a) (1 point) Classify this transformation as one of the basic types of transformations we discussed in class. Justify your answer. (b) (3 points) For what vectors w and scalars y is the following true (ignore the trivial case of w=0 or y = 0) (A– 21A)w = -p?w

Answer 1

Max2 (IR) CC 9 S= {[4] [::] - [147] [48] [:7} - - is bofiq of Sel s contain 9 vector and basis of (Max2 (IR) Contain 6 vector si s is not basis of Max2 (IR) cd 1 S- 11:3],[!!]»[::]:10:345).[!!]} Linearly in debenden ss Max (IR) and Contain u vector basis of Maxz (IR) Set larzer than any bosif 18 linearly debendent ol s is linearly dependent ce rel Ai Irm → RO theo dimension of hall space 18 losser than d may cos, may not null spare Subspace of Irm dim (Null Spare , sm ib m<d then null space than d If not farter

Munu be Ле diagond mateix H) eigen volue A is similar to A which has 1, 2, 3 with algebecic multiplicity geometric multiplicity and le 2,1 - 1, Not true because a is diagonal & A is similar to a cie A is diagonalizable A is diagonalizable and iff Algebraic multiplicin = geometric Multish 2 and Algebraic Multiplicity of 2 = Geometric multiplicity of 2= diagonalizable is not similar to diagonal Matrise us A is not 2 =) A is not Similar to Л