Test I, Math 307 Monday, January 31, 2005 1. Solve AX = B and write the answer as X = X + a X + a X + ... a X 0 1 1 2 2 r r and check your answer using A[X X X ... X ] = [B 0 0 0 ... 0]. 0 1 2 r | 4 12 12 12 20 | | x | | 16 | | 2 6 6 6 10 | | y | | 8 | | 1 1 2 3 4 | | z | | 2 | | 0 2 1 0 1 | | w | | 2 | | 1 3 3 3 5 | | u | = | 4 | --------------------------------------------------------------------------- Solution: | 4 12 12 12 20 16 | | 2 6 6 6 10 8 | | 1 1 2 3 4 2 | | 0 2 1 0 1 2 | | 1 3 3 3 5 4 | | 1 1 2 3 4 2 | | 4 12 12 12 20 16 | | 2 6 6 6 10 8 | | 0 2 1 0 1 2 | | 1 3 3 3 5 4 | | 1 1 2 3 4 2 | | 0 8 4 0 4 8 | | 0 4 2 0 2 4 | | 0 2 1 0 1 2 | | 0 2 1 0 1 2 | | 1 1 2 3 4 2 | | 0 2 1 0 1 2 | | 0 8 4 0 4 8 | | 0 4 2 0 2 4 | | 0 2 1 0 1 2 | | 1 1 2 3 4 2 | | 0 1 1/2 0 1/2 1 | | 0 8 4 0 4 8 | | 0 4 2 0 2 4 | | 0 2 1 0 1 2 | x y z=a w=b u=c rhs | 1 0 3/2 3 7/2 1 | | 0 1 1/2 0 1/2 1 | | 0 0 0 0 0 0 | | 0 0 0 0 0 0 | | 0 0 0 0 0 0 | | x | | 1 | | -3/2 | | -3 | | -7/2 | | y | | 1 | | -1/2 | | 0 | | -1/2 | | z | = | 0 | + a | 1 | +b | 0 | + c | 0 | | w | | 0 | | 0 | | 1 | | 0 | | u | | 0 | | 0 | | 0 | | 1 | Check: | 4 12 12 12 20 | | 1 -3/2 -3 -7/2 | | 16 0 0 0 | | 2 6 6 6 10 | | 1 -1/2 0 -1/2 | | 8 0 0 0 | | 1 1 2 3 4 | | 0 1 0 0 | = | 2 0 0 0 | | 0 2 1 0 1 | | 0 0 1 0 | | 2 0 0 0 | | 1 3 3 3 5 | | 0 0 0 1 | | 4 0 0 0 | It checks. 2. Find the inverse of this matrix. | 1 1 4 0 | | 0 3 4 1 | | 1 1 3 0 | | 0 2 3 1 | Solution: | 1 1 4 0 1 0 0 0 | | 0 3 4 1 0 1 0 0 | | 1 1 3 0 0 0 1 0 | | 0 2 3 1 0 0 0 1 | | 1 1 4 0 1 0 0 0 | | 0 3 4 1 0 1 0 0 | | 0 0 -1 0 -1 0 1 0 | | 0 2 3 1 0 0 0 1 | | 1 1 4 0 1 0 0 0 | | 0 1 1 0 0 1 0 -1 | | 0 0 -1 0 -1 0 1 0 | | 0 2 3 1 0 0 0 1 | | 1 0 3 0 1 -1 0 1 | | 0 1 1 0 0 1 0 -1 | | 0 0 -1 0 -1 0 1 0 | | 0 0 1 1 0 -2 0 3 | | 1 0 0 0 -2 -1 3 1 | | 0 1 0 0 -1 1 1 -1 | | 0 0 -1 0 -1 0 1 0 | | 0 0 0 1 -1 -2 1 3 | | 1 0 0 0 -2 -1 3 1 | | 0 1 0 0 -1 1 1 -1 | | 0 0 1 0 1 0 -1 0 | | 0 0 0 1 -1 -2 1 3 | -1 check | 1 1 4 0 | | -2 -1 3 1 | | 1 0 0 0 | | 0 3 4 1 | = | -1 1 1 -1 | | 0 1 0 0 | | 1 1 3 0 | | 1 0 -1 0 | | 0 0 1 0 | | 0 2 3 1 | | -1 -2 1 3 | | 0 0 0 1 | It checks. 3. (a) Write the matrix of the linear transformation of differentiation with respect to the basis 3 2x 2 2x 2x 2x x e , x e , x e , e , (b) Find some way to use the matrix from part (a) to compute the third 3 2 2x derivative of (x + 2 x + x + 1) e Solution; 3 2x 2 2x 2x 2x x e x e x e e 3 2x x e 2 3 0 0 2 2x x e 0 2 2 0 2x x e 0 0 2 1 2x e 0 0 0 2 Answer(a) | 2 0 0 0 | | 3 2 0 0 | | 0 2 2 0 | | 0 0 1 2 | f f' f'' f''' Answer(b) | 2 0 0 0 | | 1 | | 2 | | 4 | | 8 | | 3 2 0 0 | | 2 | | 7 | | 20 | | 52 | | 0 2 2 0 | | 1 | | 6 | | 26 | | 92 | | 0 0 1 2 | | 1 | | 3 | | 12 | | 50 | 3 2 2x ( 8x + 52 x + 92 x + 50 )e D[D[D[ (x^3 + 2 x^2 + x + 1) E^(2x),x],x],x] 4. Multiply these two matrices. | 1 0 1 0 0 | | 3 8 2 4 | | 0 0 0 1 1 | | 2 0 1 0 | | 0 0 3 0 0 | | 1 2 1 1 | | 1 0 0 0 1 | | 5 1 1 0 | | 1 1 1 1 1 | | 3 8 3 3 | | 1 0 1 0 0 | | 3 8 2 4 | | 4 10 3 5 | | 0 0 0 1 1 | | 2 0 1 0 | | 8 9 4 3 | | 0 0 3 0 0 | | 1 2 1 1 | = | 3 6 3 3 | | 1 0 0 0 1 | | 5 1 1 0 | | 6 16 5 7 | | 1 1 1 1 1 | | 3 8 3 3 | |14 19 8 8 | 5. Tidbits: (a) The columns of AB are linear combinations of what? The columns of A. (b) The rows of AB are linear combinations of what? The rows of B. (c) What is the nullity of a matrix? The number of columns without stairstep ones in the Row Canonical Form. (d) What is the rank of a matrix? The number of nonzero rows in the row canonical form (e) What is the relation between the row canonical form of a matrix and the invertibility of the matrix. A matrix in invertible if and only if its row canonical form is the identity matrix. (f) Write down the elementary row matrix which adds six times the first row to the third row of any 3x3 matrix and leaves the first and second rows unchanged. | 1 0 0 | | 0 1 0 | | 6 0 1 | (g) Compute 9 | Sqrt[3]/2 -1/2 | | 1/2 Sqrt[3]/2 | Answer | Cos[30] -Sin[30] | = | Cos[270] -Sin[270] | = | 0 1 | | Sin[30] Cos[30] | | Sin[270] Cos[270] | | -1 0 | (h) What are the three elementary row operations. 1. Switch two rows. 2. Multiply a row by a nonzero scalar. 3. Add a multiple of one row to another. (i) When is a function a linear transformation? f(A+B) = f(A)+f(B) f(cA) = c f(A) for all vectors A,B and all scalars c. (j) Find the dimensions of the product of these matrices A B C D 8x2 2x5 5x4 4x17 ANS will be an 8x17 matrix.