Question

(d) (4 points) Let T : R² + Rº be the transformation that rotates any vector 90 degrees counterclockwise. Let A be the standard matrix for T. Is A diagonalizable over R? What about over C? (e) (3 points) Let T : R4 → R4 be given by T(x) = Ax, A = 3 -1 7 12 0 0 0 4 0 0 5 4 0 4 2 1 Is E Im(T)? 3 (f) (9 points) Let U be a subspace of R3, and let v = 1 EU+, under the standard 2 inner product. i. Calculate projuv, the orthogonal projection of v onto U. ii. Calculate the distance from v to the closest point on U. iii. Give bounds on the dimension of U (meaning, find m and n such that m< Dim(U) <n).

Answer 1

d ) Flaxy) = (xlosgeysingo, «Singö +86595) s-y, x) (o 1) FCO)=(-1,0) T(10) = A= 1-es ChAG) = x+1 Eigenvalues of A are ti , i af Therefore Å has not real eigenvalue Hence Ais not diagonalizable work. Ais diagonalizable over's because it has two distinct eigenvalue. A= 3 oo o 14 7 54 o 12 det A &0 421 there fore Ramka z 4. Hence 7 Im = Rt. Elm (7) R 7 By definition of orthogonal projection (9 projgv = 0 Zero Vector to projuris closest point on u tou. distance is a well= √g fleq= Via

(111) may be be 0, 1, 2,6 o is subspace of on? Therefore dimension of u So 02 dim 022 dim(o) Cannot 24ut be 3. If Then dimua 3 uta 203