MAT224H1 Midterm: MAT224 MIDTERM SELF GENERATED SOLUTION 2009.pdf
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February 26, 2009: kaveh, d. raghavan, s. uppal, f. ziltener. [4] 1(a) let a be an n n matrix over r. if the characteristic polynomial of a is (x )n, false where r, then a is similar to a diagonal matrix. true (cid:18)0 1 (cid:19) But, e0 = span{ diagonalizable and therefore not similar to a diagonal matrix. has characteristic polynomial x2 (i. e. a has only the eigenvalue zero. } has dimension one = a is not. [4] 1(b) if t : u v be a linear transformation. Suppose there exists another linear transfor- mation s: v u such that for all u u , st (u) = u, and for all v v , t s(v) = v. false. T is one-to-one since if t (u1) = t (u2), then u1 = st (u1) = st (u2) = u2. T is onto since if v v , then for u = s(v),
