18.06 Linear Algebra, Final Exam Solution

(d) If the vector b is the sum of the four columns of A, write down the complete solution to Ax = b. Answer: � ⎡ � ⎡ � ⎡ 1 −2 3 � 1 ⎢ � ⎢ � ⎢ ⎢ + x2 � 1 ⎢ + x4 � −2 ⎢ x=� � 1 ⎣ � 0 ⎣ � 0 ⎣ 1 0 1 2. (11 points) This problem finds the curve y = C + D 2 t which gives the best least squares fit to the points (t, y) = (0, 6), (1, 4), (2, 0). (a) Write down the 3 equations that would be satisfied if the curve went through all 3 points. Answer: C + 1D = 6 C + 2D = 4 C + 4D = 0 (b) Find the coefficients C and D of the best curve y = C + D2 t . Answer: ⎦ T 1 1 1 � ⎤ C D � 1 1 ⎥ 1 = 14 � � 1 A A= � 1 2 4 1 � ⎦ � 1 1 1 � � AT b = � 1 2 4 Solve AT Ax̂ = AT b : ⎤ 3 7 7 21 ⎥⎤ C D ⎥ = ⎤ 10 14 ⎥ gives ⎡ ⎦ � ⎢ 3 7 ⎢= 2 ⎣ 7 21 4 ⎡ ⎦ � 6 ⎢ 10 ⎢= 4 ⎣ 14 0 ⎤ 21 −7 −7 3 ⎥⎤ 10 14 ⎥ = ⎤ 8 −2 ⎥ . (c) What values should y have at times t = 0, 1, 2 so that the best curve is y = 0? Answer: The projection is p = (0, 0, 0) if AT b = 0. In this case, b = values of y = c(2, −3, 1). 3. (11 points) Suppose Avi = bi for the vectors v1 , . . . , vn and b1 , . . . , bn in Rn . Put the v’s into the columns of V and put the b’s into the columns of B. (a) Write those equations Avi = bi in matrix form. What condition on which vectors allows A to be determined uniquely? Assuming this condition, find A from V and B. Answer: A [v1 · · · vn ] = [b1 · · · bn ] or AV = B. Then A = BV −1 if the v � s are independent. (b) Describe the column space of that matrix A in terms of the given vectors. Answer: The column space of A consists of all linear combinations of b1 , · · · , bn . (c) What additional condition on which vectors makes A an invertible matrix? Assuming this, find A−1 from V and B. Answer: If the b� s are independent, then B is invertible and A−1 = V B −1 . 4. (11 points) (a) Suppose xk is the fraction of MIT students who prefer calculus to linear algebra at year k. The remaining fraction yk = 1 − xk prefers linear algebra. At year k + 1, 1/5 of those who prefer calculus change their mind (possibly after taking 18.03). Also at year k + 1, 1/10 of those who prefer linear algebra change their mind (possibly because of this exam). � � � � � � xk+1 xk 1 k Create the matrix A to give =A and find the limit of A as k � �. yk 0 yk+1 Answer: ⎥ ⎤ .8 .1 . A= .2 .9 The eigenvector with � = 1 is ⎤ 1/3 2/3 ⎥ This is the steady state starting from . ⎤ 1 0 ⎥ . 2 of all students prefer linear algebra! I agree. 3 (b) Solve these differential equations, starting from x(0) = 1, y(0) = 0 : dx = 3x − 4y dt dy = 2x − 3y . dt Answer: ⎤ ⎥ 3 −4 A= . 2 −3 has eigenvalues �1 = 1 and �2 = −1 with eigenvectors x1 = (2, 1) and x2 = (1, 1). The initial vector (x(0), y(0)) = (1, 0) is x1 − x2 . So the solution is (x(t), y(t)) = et (2, 1) + e−t (1, 1). (c) For what initial conditions � x(0) y(0) � does the solution � x(t) y(t) � to this differential equation lie on a single straight line in R 2 for all t? Answer: If the initial conditions are a multiple of either eigenvector (2, 1) or (1, 1), the solution is at all times a multiple of that eigenvector. 5. (11 points) (a) Consider a 120� rotation around the axis x = y = z. Show that the vector i = (1, 0, 0) is rotated to the vector j = (0, 1, 0). (Similarly j is rotated to k = (0, 0, 1) and k is rotated to i.) How is j − i related to the vector (1, 1, 1) along the axis? Answer: � ⎡ −1 j−i=� 1 ⎣ 0 � ⎡ 1 is orthogonal to the axis vector � 1 ⎣ . 1 So are k − j and i − k. By symmetry the rotation takes i to j, j to k, k to i. (b) Find the matrix A that produces this rotation (so Av is the rotation of v). Explain why A3 = I. What are the eigenvalues of A? Answer: A3 = I because this is three 120� rotations (so 360� ). The eigenvalues satisfy �3 = 1 so � = 1, e2�i/3 , e−2�i/3 = e4�i/3 . (c) If a 3 by 3 matrix P projects every vector onto the plane x+2y+z = 0, find three eigenvalues and three independent eigenvectors of P. No need to compute P . Answer: The plane is perpendicular to the vector (1, 2, 1). This is an eigenvector of P with � = 0. The vectors (−2, 1, 0) and (1, −1, 1) are eigenvectors with � = 0. 6. (11 points) This problem is about the matrix � ⎡ 1 2 A = � 2 4 ⎣. 3 6 (a) Find the eigenvalues of AT A and also of AAT . For both matrices find a complete set of orthonormal eigenvectors. Answer: � ⎡ ⎤ ⎥ 1 2 1 2 3 � 14 28 T A A= 2 4 ⎣= 2 4 6 28 56 3 6 ⎤ ⎥ ⎤ ⎥ 1 1 1 −2 and x2 = � . has �1 = 70 and �2 = 0 with eigenvectors x1 = � 1 5 2 5 ⎥ ⎤ � ⎡ � ⎥ 1 2 ⎤ 5 1 2 3 T � ⎣ � AA = 2 4 = 10 2 4 6 3 6 15 � ⎡ � 1 −2 1 � ⎣ 1 � x1 = � and x2 = � 1 2 14 3 5 0 ⎡ 10 15 20 30 ⎣ has �1 = 70, �2 = 0, �3 = 0 with 30 45 ⎡ � ⎡ 3 ⎣ and x3 = �1 � 6 ⎣ . 70 −5 (b) If you apply the Gram-Schmidt process (orthonormalization) to the columns of this matrix A, what is the resulting output? Answer: Gram-Schmidt will find the unit vector � ⎡ 1 1 q1 = � � 2 ⎣ . 14 3 But the construction of q2 fails because column 2 = 2 (column 1). (c) If A is any m by n matrix with m > n, tell me why AA T cannot be positive definite. Is AT A always positive definite? (If not, what is the test on A?) Answer AAT is m by m but its rank is not greater than n (all columns of AAT are combinations of columns of A). Since n < m, AAT is singular. AT A is positive definite if A has full colum rank n. (Not always true, A can even be a zero matrix.) 7. (11 points) This problem is to find the determinants of � ⎡ � ⎡ � 1 1 1 1 0 1 1 1 x 1 � 1 1 1 1 ⎢ � 1 1 1 1 ⎢ � 1 1 ⎢ ⎢ � � A=� � 1 1 1 0 ⎣ B=� 1 1 1 0 ⎣ C=� 1 1 1 1 0 0 1 1 0 0 1 1 1 1 1 0 ⎡ 1 1 ⎢ ⎢ 0 ⎣ 0 (a) Find det A and give a reason. Answer: det A = 0 because two rows are equal. (b) Find the cofactor C11 and then find det B. This is the volume of what region in R 4 ? Answer: The cofactor C11 = −1. Then det B = det A − C11 = 1. This is the volume of a box in R4 with edges = rows of B. (c) Find det C for any value of x. You could use linearity in row 1. Answer: det C = xC11 + det B = −x + 1. Check this answer (zero), for x = 1 when C = A. 8. (11 points) (a) When A is similar to B = M −1 AM, prove this statement: If Ak � 0 when k � �, then also B k � 0. Answer: A and B have the same eigenvalues. If Ak � 0 then all |�| < 1. Therefore B k � 0. (b) Suppose S is a fixed invertible 3 by 3 matrix. This question is about all the matrices A that are diagonalized by S, so that S −1 AS is diagonal. Show that these matrices A form a subspace of 3 by 3 matrix space. (Test the requirements for a subspace.) Answer: If A1 and A2 are in the space, they are diagonalized by S. Then S −1 (cA1 + dA2 )S is diagonal + diagonal = diagonal. (c) Give a basis for the space of 3 by 3 diagonal matrices. Find a basis for the space in part (b) — all the matrices A that are diagonalized by S. Answer: A basis for the diagonal matrices is � ⎡ � ⎡ � ⎡ 1 0 0 ⎣ D2 = � ⎣ D3 = � ⎣ 0 1 0 D1 = � 0 0 1 Then SD1 S −1 , SD2 S −1 , SD3 S −1 are all diagonalized by S: a basis for the subspace. (b) Compute the matrix AT A. What is its rank? What is its nullspace? Answer: � � AT A = � � has rank 3 like A. The nullspace ⎡ 3 −1 −1 −1 −1 3 −1 −1 ⎢ ⎢ −1 −1 3 −1 ⎣ −1 −1 −1 3 is the line through (1, 1, 1, 1). (c) Suppose v1 = 1 and v4 = 0. If each edge contains a unit resistor, the currents (w 1 , w2 , w3 , w4 , w5 , w6 ) on the 6 edges will be w = −Av by Ohm’s Law. Then Kirchhoff’s Current Law (flow in = flow out at every node) gives AT w = 0 which means AT Av = 0. Solve AT Av = 0 for the unknown voltages v2 and v3 . Find all 6 currents w1 to w6 . How much current enters node 4? Answer: Note: As stated there is no solution (my apologies!). All solutions to AT Av = 0 are multiples of (1, 1, 1, 1) which rules out v1 = 1 and v4 = 0. Intended problem: I meant to solve the reduced equations using KCL only at nodes 2 and 3. In fact symmetry gives v2 = v3 = 12 . Then the currents are w1 = w2 = w5 = w6 = 1 2 around the sides and w3 = 1 and w4 = 0 (symmetry). So w3 + w5 + w6 = the total current into node 4. 1 2 is