-3 6 8. Let ū = = and w : Is it possible to find a scalar t such 1 -3 -4 that ū + tỉ is parallel to w.

Let v = (60, – 5, – 5) and w = (– 4, – 30, 22) be ordered triples in R’. Find two scalars c1 and c2 such that C1V + c2w = (44, – 20, 8). - List the exact values of c1 and c2 separated by commas:

43. Linear combination Let u = (1, 2, 1), v = (1,-1,-1), w = (1, 1,-1), and z = (2, -3,-4). Find scalars a, b, and c such that z = au + bv + cw.

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