y 13. T y | B1 %3D B2: %3D 1

Let A = [1 -1 -1              -1   1 -1              -1  -1  1]. Find a diagonalization if possible.

Find the matrix A of the linear transformation T from R2 to R² that rotates any vector through an angle of 45° in the clockwise direction. A = 9-

Let B = {b,.b2} be bases for a vector space V, and suppose b, = 8c, - 2c2 and b2 = 9c, - 3c2. and C = %D a. Find the change-of-coordinates matrix from B to C. b. Find [x]c for x = - 3b, + 6b2. Use part (a). ... = C-B a.

Find the matrix A of the linear transformation T from R2 to R² that rotates any vector through an angle of 150° in the clockwise direction. A =

Find the transpose, conjugate, and adjoint of (i) Transpose is idempotent: (4¹) = A. (ii) Transpose respects addition: (A + B) = A¹+ B¹. (iii) Transpose respects scalar multiplication: (c. A)¹ = c A. These three operations are defined even when m‡n. The transpose and adjoint are both functions from CX to Cxm These operations satisfy the following properties for all c E C and for all A, B € CX". (iv) Conjugate is idempotent: A = A. (v) Conjugate respects addition: A 6-3i 0 1 B = A + B (vi) Conjugate respects scalar multiplication: C. A = c. A. (vii) Adjoint is idempotent: (4†)t = 4. (viii) Adjoint respects addition: (A + B) =A+B+. (ix) Adjoint relates to scalar multiplication: (CA)t = c · At 2 + 12i 5+2.1i 2+5i -19i 17 3-4.5i

Let T : R3 → R3 be the orthogonal projection onto the line L in R3 containing all scalar multiples of the vector 1 (a) Express T in matrix form using a 3x3 matrix A. (b) Find the rank of the matrix A.