Let M2x2 be the vector space of all 2 x 2 matrices, and define -[: :} T: M2x2 → M2x2 by T(A) = A + AT , where A = a. Show that T is a linear transformation. b. Let B be any element of M2x2 such that BT = B. Find an A in M2x2 such that T (A) = B. c. Show that the range of T is the set of B in M2x2 with the property that BT = B. d. Describe the kernel of T.

Let B= (b,, b,, b,} and D= (d,, d,} be bases for vector spaces V and W, respectively. Let T :V- W be a linear transformation with the following properties. T(b,) = - 3d, +2d2, T(b2) = - 4d, + 4d2, T(b3) = 5d2 Find the matrix for T relative to B and D. The matrix for T relative to B and D isO.

Let M₂x2 be the vector space of all 2 x 2 matrices and define a linear transformation T: M2x2 → M2x2 by a [id] Describe the kernel of T. C T(A) = A + AT, where A = O No answer text provided. odlo 0 -a ° 0 O [1] :d is a real number} [9] : b is a real number} -b O (1 :d is a real number} (d c is a real number} °41 :d is a real number} : a,b,c,d are real numbers} [9] : b is a real number}