MATH 223 Assignment #3 due Friday September 26. I like to see the solutions to a system of equations in Parametric Vector Form (or Vector Parametric Form if you wish). If we find a set of solutions: x1 x2 x3 x4 x5 x6 = −3r = r = = = = 1/3 −4s −2t −2s s t for all choices r, s, t ∈ R then we can write the set of solutions in parametric vector form as follows: −2 −4 −3 0 x1 0 0 1 0 x2 x3 = 0 + r 0 + s −2 + t 0 , 0 1 0 0 x 4 1 0 0 0 x5 0 0 0 1/3 x6 r, s, t ∈ R 1. Give the solutions in vector parametric form for the two planes: a) π1 = {(x, y, z) : 2x − 2y + 3z = 5}. b) π2 = {(x, y, z) : 3y = 6}. 2. Give the vector parametric form of all solutions to the following system of equations: 2x1 2x1 3x1 3x1 +4x4 +5x4 +8x4 +13x4 +2x2 +4x2 +6x5 +7x5 +9x5 +12x5 = = = = 14 16 27 39 3. Express the line given by the vectors 1 1 x y = 1 + s 2 , z 1 −1 s∈R as the set of solutions to a system of equations in x, y, z (two equations suffice). Then use Gaussian Elimination on this system to re-express the solutions in vector parametric form. 4. Express the solutions to x1 0 1 0 1 3 7 x2 0 0 0 1 2 x3 = 5 0 0 0 0 0 x4 0 x5 in vector parametric form as x = a + sb + tc + ud. Show that the only solution to s [bcd] t = 0 u is s t = 0. u 5. Express the inverse of the following matrix A as a product of elementary matrices and use this product to express the matrix A as a product of elementary matrices. 1 1 0 A = 2 3 1 0 0 1 6. Let A be a 2×2 matrix with first column x and second column y. We wish to show that | det(A)| is the area of the parallelogram formed by the two vectors x, y. Using assignment 2, question 1 (a), we readily deduce that det(A) = 0 if and only if the parallelogram is degenerate with no area. So assume det(A) 6= 0. a) There exists an angle θ such that R(θ)x = x0 points in the direction of the x-axis. Let y 0 = R(θ)y. Show that the area of the parallelogram formed by x0 , y0 is the same as the area of the parallelogram formed by the two vectors x, y. # " 1 s b) We can choose a value s so that if we set S = , then this shear matrix has Sx0 = x0 0 1 while Sy0 = y00 has y00 in the direction of the y-axis or its opposite. Show that the area of the parallelogram formed by x0 , y00 is the same as the area of the parallelogram formed by the two vectors x0 , y0 . c) If we let B be a 2 × 2 matrix with first column x0 and second column y00 , show that | det(B)| is the area of the rectangular box formed by x0 , y00 . d) Using the product rule for determinants and verifying that det(R(θ)) = 1 and det(S) = 1, show that | det(A)| is the area of the parallelogram formed by the two vectors x, y. 7. Show that the 2×2 matrix of 0’s is diagonalizable (I mention this because many students confuse invertibility and diagonalizability; they are not related). 8. This is harder than question 7 from assignment 2 and essentially is the reverse observation. Let A be a 2 × 2 matrix (not necessarily diagonalizable) and define Assume limn→∞ xn yn 1 xn = An . 2 yn 1 is an eigenvector of A. 1 = 1. Show that 1 , 1 Hint: This is one possible approach. Rewrite the matrix with respect to the two vectors 1 .. Explain why we can find c1 , c2 , c3 , c4 with 2 1 1 1 A = c1 + c2 ; 1 1 2 1 1 1 A = c3 + c4 . 2 1 2 1 1 What do you need to show about c1 , c2 , c3 , c4 so that An 1 1 1 with An+1 = A An 2 2 2 . is an eigenvector of A? Now compare