Solutions to problems from section 1.5 6. Write the solution set of the given homogeneous system in parametric vector form. x1 + 3x2 5x3 = 0 x1 + 4x2 8x3 = 0 3x1 7x2 + 9x3 = 0 Solution: First we nd the general solution: 1 3 5 0 1 3 5 0 1 3 5 0 N R2 =R2 R1
Solutions to even numbered problems from section 1.6 4a. Suppose an economy has four sectors, Agriculture (A), Energy (E), Manufacturing (M), And Transportation (T). Sector A sells 10% of its output to E and 25% to M and retains the rest. Sector E sells 3
Solutions to problems from section 1.9 For questions 2, 4, 10 assume that T is a linear map and nd the standard matrix A for T . [Recall that by theorem 10 we know the columns of A will be the images of the ei s under T .] 2. T : R3 R2 , T (e1 ) = Solutio
Solutions to problems from section 2.1 Throughout this assignment you were to assume the each matrix expression is dened. 4. Compute A 5I3 and (5I3 )A when 9 1 3 A = 8 7 6 . 4 1 8 Solution: 4 1 3 9 1 3 500 A 5I3 = 8 2 6 . 7 6 0 5 0 = 8 005 4 1 3 4 1 8 45
imA= ColA
imA= orthogonal complement of imA=ker AT
Rank- Nullity Theorem: Suppose A is an n m matrix. Then rank A + nullity A = m
Least Square Method: Ax=b ATAx=ATb solve for x projection (m, b)
When is Ax=b unique? The Nul of ATA
Characteristic Equation: