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Name Math 311 ID Final Exam Spring 2002 Section 503 P. Yasskin x 1. 10 points Extra Credit Determine if and where the line intersects the plane y z 2 = y = −8 3 1 +r −1 2 /10 3 /30 3 /40 4 /30 1 z x 1 EC 2 3 4 +t 5 6 3 +s 6 . 4 1 2 1 2. 40 points Let Mp, q be the vector space of p × q matrices, and let P = 4 2 . 6 3 Consider the linear function L : M2, 2 a. 5 Let X = a b c d M3, 2 given by LX = PX and compute LX . b. 2 Identify the domain of L. What is its dimension? c. 2 Identify the codomain of L. What is its dimension? d. 8 Identify the kernel (null space) of L. Give a basis and the dimension. 2 Problem 2 continued: e. 8 Identify the image (range) of L. Give a basis and the dimension. f. 3 Is L onto? Why? g. 3 Is L one-to-one? Why? h. 2 Check that the dimensions of the kernel and image are consistent with the dimensions of the domain and codomain. 3 Problem 2 continued: i. 7 Find the matrix of L relative to the bases E1 = 1 0 0 0 E2 = 0 1 0 0 1 0 F1 = F4 = 0 0 E3 = 0 0 1 0 0 0 F2 = 1 0 E4 = F3 = 0 0 0 0 1 0 0 1 0 0 0 0 0 0 F5 = 0 1 0 0 0 1 0 0 0 0 0 0 0 0 F6 = 0 0 0 1 4 3. 30 points On the vector space P 2 = polynomials given by polynomials of degree less than 2 ⟨p, q ⟩ a. 15 Show ⟨p, q ⟩ consider the function of two ∞ = ∫ px qx e −x dx 0 is an inner product. b. 15 Find the angle θ between the polynomials px = 1 and qx = x. You may use these integrals without proof: ∞ ∞ ∫ e −x dx = 1, ∫ xe −x dx = 1, 0 0 ∞ ∫ x 2 e −x dx = 2, 0 ∞ ∫ x 3 e −x dx = 6 0 5 4. 30 points Gauss’ Theorem states that if V is a volume in R 3 and ∂V is its boundary surface oriented ⃗ is a nice vector field on V then outward from V and F ⃗ dV = ∫∫ F ⃗ ⋅ dS ∫∫∫ ⃗∇ ⋅ F V ∂V ⃗ = xz 2 , −yz 2 , x 2 z + y 2 z Verify Gauss’ Theorem if F and V is the volume above the paraboloid z = x 2 + y 2 and below the plane z = 9. Notice that ∂V consists of the paraboloid P and a disk D. Be sure to use the correct orientations for P and D. ⃗ dV : a. 6 Compute ∫∫∫ ⃗ ∇⋅F V ⃗= ∇⋅F i. ⃗ ⃗ dV = ii. ∫∫∫ ⃗ ∇⋅F V 6 Problem 4 continued: ⃗ ⋅ dS : b. 10 For the paraboloid P compute ∫∫ F ⃗ = xz 2 , −yz 2 , x 2 z + y 2 z . Recall F P ⃗ r, θ = i. R ⃗r = ii. R ⃗θ = iii. R ⃗= iv. N ⃗ R ⃗ r, θ v. F ⃗ R ⃗ r, θ vi. F = ⋅ ⃗= N ⃗ ⋅ dS = vii. ∫∫ F P 7 Problem 4 continued: ⃗ ⋅ dS : c. 10 For the disk D compute ∫∫ F ⃗ = xz 2 , −yz 2 , x 2 z + y 2 z . Recall F D ⃗ r, θ = i. R ⃗r = ii. R ⃗θ = iii. R ⃗= iv. N ⃗ R ⃗ r, θ v. F ⃗ R ⃗ r, θ vi. F = ⋅ ⃗= N ⃗ ⋅ dS = vii. ∫∫ F D d. 4 Verify the two sides of Gauss’ Theorem are equal. 8