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Math 333: Homework 5: Due: Tuesday, Dec. 1, 2015 in class 1. [5pt] A basis for V = IR3 is S = {u1 , u2 , u3 } = {(2, 1, 0), (1, 1, 1), (4, 1, 1)} Use the Gram-Schmidt process to derive an orthogonal basis S? . 2. [5pt] Let V = IR3 and W = span {u1 , u2 } = span {(1, 1, 1), (0, 1, 1)} be a subspace defined by its orthogonal basis vectors uk . Find a matrix A 2 IR3⇥3 such that T (v) ⌘ Av = projW v An example of this procedure is given in the posted notes. 3. [5pt] We are given the data points Z = {(x1 , y1 ), (x2 , y2 ), (x3 , y3 )} = {( 1, 2), (0, 1), (1, 3)} and want a least squares fit to the quadratic model y = ax2 + c Thus, x = (a, c) must be the least squares solution of Ax = b where 2 x21 A = 4 x22 x23 3 1 1 5 1 2 3 y1 b = 4 y2 5 y3 Find x and plot the data along with y = ax2 + c. 4. [5pt] Let x̂ be the least squares solution of Ax = b and define the residual vector r ⌘ Ax̂ b. Prove k r k2 = < b, r >. To do so, start noting k r k2 = (Ax̂ b)T (Ax̂ b) · · · Write this out as a formal proof explaining each step. 5. [5pt] Let T : X ! IR3 be a linear transformation with basis S = {u1 , u2 , u3 }. Given 0 1 0 1 0 1 1 5 1 T (u1 ) = @ 2 A T (u2 ) = @ 7 A T (u3 ) = @ 1 A 1 1 1 Find a basis for the range R(T ). DO NOT put T (uk ) in the rows of any matrix. 1 6. [25pt] In each the the four problems below a linear transformation T : X ! Y is defined. For each, clearly define the kernel ker(T ) and the range R(T ) of each transformation. When T has an inverse, define its formula and its domain. a) T : P5 ! IR (average) T (u) = b) T : P2 ! P2 Z 1 u(x)dx 1 T (u) = u(x + 1) u(x c) T : M22 ! IR T (u) = trace(u) , d) T : P2 ! P2 T (u) = x e) T : P2 ! P2 u= du dx T (u) = u(2x + 1) 2 1) a c b d