MAT 242 Test 3 SOLUTIONS, FORM A

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1 MAT Test SOLUTIONS, FORM A. Let v =, v =, and v =. Note that B = { v, v, v } is an orthogonal set. Also, let W be the subspace spanned by { v, v, v }. A = 8 a. [5 points] Find the orthogonal projection of into W, without inverting any matrices or 9 solving any systems of linear equations. c = v u v v = 75 5 =, c = v u v v = 5 5 =, c = v u = 75 v v 5 =, 6 p = A c = Grading: + points for finding the c i s, +5 points for finding p. Grading for common mistakes: 5 points for using the (A A) A u formula. b. [ points] Find an orthonormal basis for W. /5 /5 v v = 5 v v = 5 v v = 5 /5 /5 /5 /5 /5 /5,, /5 /5 /5 /5 Grading: +5 points for knowing to divide a vector by its length, +5 points for doing it with the orthogonal basis. Grading for common mistakes: +5 points (total) for doing Gram-Schmidt.

2 MAT Test SOLUTIONS, FORM A 5. Let W be the subspace spanned by,,. Note that this basis is not orthogonal a. [5 points] Find the vector in W closest to. 5 c = ( A A ) ( A B ) = 5 7 p = A c = 5 Grading: + points for finding A and u, + points for the formula, + points for the calculation. Grading for common mistakes: 5 points for only finding the coordinates. b. [5 points] Find an orthogonal basis for W. NEW = OLD = NEW = OLD OLD NEW NEW = 9 = NEW NEW 9 NEW = OLD OLD NEW NEW OLD NEW 5 NEW = 7 9 = NEW NEW NEW NEW Grading: +5 points for each vector. Grading for common mistakes: +5 points (total) for dividing each vector by its length.

3 MAT Test SOLUTIONS, FORM A. Do the following, for the following set of data points: (, 9), (, ), (, ), (, 7). a. [ points] Find the parabola y = ax + bx + c which best fits these points B = 7 a b = (A A) (A B) = 9/979 5/979 c 6/979 y = x x = (.) x + (.7585) x Grading: + points for A, + points for B, + points for finding the coefficients, + points for the equation y =. b. [ points] Find the parabola y = ax + c with no linear term which best fits these points B = 7 a = (A A) (A B) = c y = x = (.) x. Grading: + points for A, + points for B, + points for finding the coefficients, + points for the equation y =.

4 MAT Test SOLUTIONS, FORM A. [ points] Find the Least Squares Solution to the following system of linear equations: x + 5x x = x + x x = x + x x = x x + 5x = ˆx = ( A A ) ( A B ) = /7 /7 = / Grading: +5 points for writing down A and B, + points for the normal equation, + points for the calculation. Grading for common mistakes: points for finding Aˆx. 5. [5 points] Find a basis for W, the orthogonal complement of W, if W is the subspace spanned by Parameterized by: A = [ ] RREF [ / ] x + x x x = x / x = α + β + γ x x Grading: +5 points for A, + points the null space basis. Grading for common mistakes: +5 points (total) for Gram-Schmidt.

5 MAT Test SOLUTIONS, FORM B. Let v =, v =, and v =. Note that B = { v, v, v } is an orthogonal set. Also, let W be the subspace spanned by { v, v, v }. 5 a. [5 points] Find the vector in W closest to, without inverting any matrices or solving any 7 systems of linear equations. c = v u = 7 =, v v 9 c = v u v v = 9 =, c = v u = v v =, 6 p = A c = 6 Grading: + points for finding the c i s, +5 points for finding p. Grading for common mistakes: 5 points for using the (A A) A u formula. b. [ points] Find an orthonormal basis for W. v v = 9 v v = 9 v v = / / / /,, / / Grading: +5 points for knowing to divide a vector by its length, +5 points for doing it with the orthogonal basis. Grading for common mistakes: +5 points (total) for doing Gram-Schmidt.

6 MAT Test SOLUTIONS, FORM B Let W be the subspace spanned by,,. Note that this basis is not orthogonal a. [5 points] Find the vector in W closest to. 8 9 c = ( A A ) ( A B ) = 8 p = A c = 6 8 Grading: + points for finding A and u, + points for the formula, + points for the calculation. Grading for common mistakes: 5 points for only finding the coordinates. b. [5 points] Find an orthogonal basis for W. NEW = OLD = NEW = OLD OLD NEW 5 NEW = 5 = NEW NEW NEW = OLD OLD NEW NEW NEW NEW OLD NEW NEW NEW NEW = Grading: +5 points for each vector. Grading for common mistakes: +5 points (total) for dividing each vector by its length. =

7 MAT Test SOLUTIONS, FORM B. Do the following, for the following set of data points: ( 5, ), (, 7), (, ), (, 7). a. [ points] Find the parabola y = ax + bx + c which best fits these points B = 7 a b = (A A) (A B) = 8/78 6/78 c 57/78 y = 8 78 x x = (.66) x + ( ) x Grading: + points for A, + points for B, + points for finding the coefficients, + points for the equation y =. b. [ points] Find the parabola y = ax + c with no linear term which best fits these points B = 7 a = (A A) (A B) = c 968/7 95/7 y = x = ( ) x Grading: + points for A, + points for B, + points for finding the coefficients, + points for the equation y =.

8 MAT Test SOLUTIONS, FORM B. [ points] Find the Least Squares Solution to the following system of linear equations: x + 5x = x + x + x = x = x + x + x = 7 ˆx = ( A A ) ( A B ) = 5/87 = /97 589/97 Grading: +5 points for writing down A and B, + points for the normal equation, + points for the calculation. Grading for common mistakes: points for finding Aˆx. 5. [5 points] Find a basis for W, the orthogonal complement of W, if W is the subspace spanned by A = [ ] RREF [ ] x x + x = Parameterized by: x x = α + β + γ x x Grading: +5 points for A, + points the null space basis. Grading for common mistakes: +5 points (total) for Gram-Schmidt.

9 MAT Test SOLUTIONS, FORM C. Let v =, v =, and v =. Note that B = { v, v, v } is an orthogonal set. Also, let W be the subspace spanned by { v, v, v }. a. [5 points] Find the vector in W closest to, without inverting any matrices or solving any systems of linear equations. c = v u v v = 8 9 =, c = v u v v = 7 9 =, c = v u v v = =, p = A c = Grading: + points for finding the c i s, +5 points for finding p. Grading for common mistakes: 5 points for using the (A A) A u formula. b. [ points] Find an orthonormal basis for W. v v = 9 v v = 9 v v = / /,, / / / / Grading: +5 points for knowing to divide a vector by its length, +5 points for doing it with the orthogonal basis. Grading for common mistakes: +5 points (total) for doing Gram-Schmidt.

10 MAT Test SOLUTIONS, FORM C 9. Let W be the subspace spanned by,,. Note that this basis is not orthogonal. a. [5 points] Find the orthogonal projection of into W. c = ( A A ) ( A B ) = p = A c = Grading: + points for finding A and u, + points for the formula, + points for the calculation. Grading for common mistakes: 5 points for only finding the coordinates. b. [5 points] Find an orthogonal basis for W. NEW = OLD = NEW = OLD OLD NEW NEW = 9 = NEW NEW 9 9 NEW = OLD OLD NEW NEW OLD NEW NEW = 9 7 = NEW NEW NEW NEW 9 9 Grading: +5 points for each vector. Grading for common mistakes: +5 points (total) for dividing each vector by its length.

11 MAT Test SOLUTIONS, FORM C. Do the following, for the following set of data points: ( 5, ), (, ), (, ), (, 8). a. [ points] Find the line y = ax + b which best fits these points. 5 B = 8 a = (A A) (A B) = b 9/ 66/ y = 9 x + 66 = (.98) x Grading: + points for A, + points for B, + points for finding the coefficients, + points for the equation y =. b. [ points] Find the parabola y = ax + c with no linear term which best fits these points. 5 6 B = 8 a = (A A) (A B) = c 8/56 577/5 y = 8 56 x = (.59987) x Grading: + points for A, + points for B, + points for finding the coefficients, + points for the equation y =.

12 MAT Test SOLUTIONS, FORM C. [ points] Find the Least Squares Solution to the following system of linear equations: x + x + x = 5 x + x + 5x = x + 5x + x = x + x = 5 ˆx = ( A A ) ( A B ) = 785/99 9/99 = /.8959 Grading: +5 points for writing down A and B, + points for the normal equation, + points for the calculation. Grading for common mistakes: points for finding Aˆx. 5. [5 points] Find a basis for W, the orthogonal complement of W, if W is the subspace spanned by,, Parameterized by: A = RREF / x + x x + x = x + x x = x + x x + x = x / x = α x x Grading: +5 points for A, + points for the null space basis. Grading for common mistakes: +5 points (total) for Gram-Schmidt.

13 MAT Test SOLUTIONS, FORM D. Let v =, v =, and v =. Note that B = { v, v, v } is an orthogonal set. Also, let W be the subspace spanned by { v, v, v }. A = a. [5 points] Find the orthogonal projection of into W, without inverting any matrices or solving any systems of linear equations. c = v u v v = =, c = v u v v = 9 9 =, c = v u = v v 9 =, p = A c = Grading: + points for finding the c i s, +5 points for finding p. Grading for common mistakes: 5 points for using the (A A) A u formula. b. [ points] Find an orthonormal basis for W. v v = v v = 9 v v = 9 / / / /,, / / Grading: +5 points for knowing to divide a vector by its length, +5 points for doing it with the orthogonal basis. Grading for common mistakes: +5 points (total) for doing Gram-Schmidt.

14 MAT Test SOLUTIONS, FORM D 7. Let W be the subspace spanned by,,. Note that this basis is not orthogonal. 6 a. [5 points] Find the vector in W closest to. 7 9 c = ( A A ) ( A B ) = p = A c = 9 8 Grading: + points for finding A and u, + points for the formula, + points for the calculation. Grading for common mistakes: 5 points for only finding the coordinates. b. [5 points] Find an orthogonal basis for W. NEW = OLD = 7 NEW = OLD OLD NEW NEW = 75 = NEW NEW = 5 5 NEW = OLD OLD NEW NEW NEW NEW OLD NEW NEW NEW NEW = Grading: +5 points for each vector. Grading for common mistakes: +5 points (total) for dividing each vector by its length.

15 MAT Test SOLUTIONS, FORM D. Do the following, for the following set of data points: (, 5), (, ), (, ), (, ). a. [ points] Find the line y = ax + b which best fits these points. 5 B = a = (A A) (A B) = b 8/5 56/5 y = 8 5 x + 56 = (.8696) x Grading: + points for A, + points for B, + points for finding the coefficients, + points for the equation y =. b. [ points] Find the parabola y = ax + c with no linear term which best fits these points B = a = (A A) (A B) = c /57 87/57 y = 57 x = (.896) x +.9 Grading: + points for A, + points for B, + points for finding the coefficients, + points for the equation y =.

16 MAT Test SOLUTIONS, FORM D. [ points] Find the Least Squares Solution to the following system of linear equations: x x + 5x = 7 x + x x = 5 x + x x = x 5x = ˆx = ( A A ) ( A B ) 67/ = 96/ = /.99 Grading: +5 points for writing down A and B, + points for the normal equation, + points for the calculation. Grading for common mistakes: points for finding Aˆx. 5. [5 points] Find a basis for W, the orthogonal complement of W, if W is the subspace spanned by, Parameterized by: A = RREF x x = x + x + x x = / / x / x / = α + β x x Grading: +5 points for A, + points the null space basis. Grading for common mistakes: +5 points (total) for Gram-Schmidt.

17 MAT Test SOLUTIONS, FORM E. Let v =, v =, and v =. Note that B = { v, v, v } is an orthogonal set. Also, let W be the subspace spanned by { v, v, v }. a. [5 points] Find the vector in W closest to, without inverting any matrices or solving any systems of linear equations. c = v u v v = =, c = v u v v = =, c = v u = v v =, p = A c = Grading: + points for finding the c i s, +5 points for finding p. Grading for common mistakes: 5 points for using the (A A) A u formula. b. [ points] Find an orthonormal basis for W. v v = v v = v v =,, Grading: +5 points for knowing to divide a vector by its length, +5 points for doing it with the orthogonal basis. Grading for common mistakes: +5 points (total) for doing Gram-Schmidt.

18 MAT Test SOLUTIONS, FORM E. Let W be the subspace spanned by,,. Note that this basis is not orthogonal. 5 a. [5 points] Find the orthogonal projection of into W. 5 c = ( A A ) ( A B ) = 6 p = A c = 5 Grading: + points for finding A and u, + points for the formula, + points for the calculation. Grading for common mistakes: 5 points for only finding the coordinates. b. [5 points] Find an orthogonal basis for W. NEW = OLD = NEW = OLD OLD NEW NEW = 9 = NEW NEW 9 NEW = OLD OLD NEW NEW OLD NEW NEW = 7 8 NEW NEW NEW NEW = Grading: +5 points for each vector. Grading for common mistakes: +5 points (total) for dividing each vector by its length.

19 MAT Test SOLUTIONS, FORM E. Do the following, for the following set of data points: (, ), (, ), (, ), (, 6). a. [ points] Find the parabola y = ax + bx + c which best fits these points. 6 B = 6 a b = (A A) (A B) = c 9/ 9/6 95/6 y = 9 x x 95 6 = (9.5) x + ( ) x.6586 Grading: + points for A, + points for B, + points for finding the coefficients, + points for the equation y =. b. [ points] Find the parabola y = ax + bx passing through the origin which best fits these points. 6 B = 6 a = (A A) (A B) = b 957/ 6/ y = 957 x + 6 x = (9.788) x + ( 5.995) x Grading: + points for A, + points for B, + points for finding the coefficients, + points for the equation y =.

20 MAT Test SOLUTIONS, FORM E. [ points] Find the Least Squares Solution to the following system of linear equations: x + x = 5 x + 5x = 7 5x + x x = x x = ˆx = ( A A ) ( A B ) 8/ = 85/68 = / Grading: +5 points for writing down A and B, + points for the normal equation, + points for the calculation. Grading for common mistakes: points for finding Aˆx. 5. [5 points] Find a basis for W, the orthogonal complement of W, if W is the subspace spanned by Parameterized by: A = [ ] RREF [ ] x x + x = x x = α + β + γ x x Grading: +5 points for A, + points the null space basis. Grading for common mistakes: +5 points (total) for Gram-Schmidt.

21 MAT Test SOLUTIONS, FORM F. Let v =, v =, and v =. Note that B = { v, v, v } is an orthogonal set. Also, let W be the subspace spanned by { v, v, v }. a. [5 points] Find the orthogonal projection of into W, without inverting any matrices or solving any systems of linear equations. c = v u v v = =, c = v u v v = =, c = v u v v = =, p = A c = Grading: + points for finding the c i s, +5 points for finding p. Grading for common mistakes: 5 points for using the (A A) A u formula. b. [ points] Find an orthonormal basis for W. v v = v v = v v =,, Grading: +5 points for knowing to divide a vector by its length, +5 points for doing it with the orthogonal basis. Grading for common mistakes: +5 points (total) for doing Gram-Schmidt.

22 MAT Test SOLUTIONS, FORM F 7. Let W be the subspace spanned by,,. Note that this basis is not orthogonal. 7 a. [5 points] Find the vector in W closest to. 9 6 c = ( A A ) ( A B ) = 8 p = A c = 9 6 Grading: + points for finding A and u, + points for the formula, + points for the calculation. Grading for common mistakes: 5 points for only finding the coordinates. b. [5 points] Find an orthogonal basis for W. NEW = OLD = NEW = OLD OLD NEW NEW = 6 = NEW NEW NEW = OLD OLD NEW NEW OLD NEW NEW = 9 8 = NEW NEW NEW NEW 9 9 Grading: +5 points for each vector. Grading for common mistakes: +5 points (total) for dividing each vector by its length.

23 MAT Test SOLUTIONS, FORM F. Do the following, for the following set of data points: ( 5, ), (, 6), (, 5), (, 79). a. [ points] Find the parabola y = ax + bx + c which best fits these points B = 5 79 a b = (A A) (A B) = c 55/78 58/78 95/78 y = x + 58 x = (.7867) x + ( ) x.998 Grading: + points for A, + points for B, + points for finding the coefficients, + points for the equation y =. b. [ points] Find the parabola y = ax + bx passing through the origin which best fits these points B = 5 79 a = (A A) (A B) = b [ /6 66/6 y = 6 x x = (.755) x + ( 8.75) x Grading: + points for A, + points for B, + points for finding the coefficients, + points for the equation y =. ]

24 MAT Test SOLUTIONS, FORM F. [ points] Find the Least Squares Solution to the following system of linear equations: 5x x x = 5 x + x x = 7 5x + 5x x = 6 x x + x = 6 ˆx = ( A A ) ( A B ) 89/77 = 6779/875 = /98.89 Grading: +5 points for writing down A and B, + points for the normal equation, + points for the calculation. Grading for common mistakes: points for finding Aˆx. 5. [5 points] Find a basis for W, the orthogonal complement of W, if W is the subspace spanned by A = [ ] RREF [ ] x + x + x = Parameterized by: x x = α + β + γ x x Grading: +5 points for A, + points the null space basis. Grading for common mistakes: +5 points (total) for Gram-Schmidt.

25 MAT Test SOLUTIONS, FORM MAKE-UP. Let v =, v =, and v =. Note that B = { v, v, v } is an orthogonal set. Also, let W be the subspace spanned by { v, v, v }. A = 7 a. [5 points] Find the vector in W closest to, without inverting any matrices or solving any 6 systems of linear equations. c = v u v v = 5 5 =, c = v u v v = 75 5 =, c = v u v v = 5 5 =, p = A c = Grading: + points for finding the c i s, +5 points for finding p. Grading for common mistakes: 5 points for using the (A A) A u formula. b. [ points] Find an orthonormal basis for W. v v = 5 v v = 5 v v = 5 /5 /5 /5 /5 /5 /5,, /5 /5 /5 /5 /5 /5 Grading: +5 points for knowing to divide a vector by its length, +5 points for doing it with the orthogonal basis. Grading for common mistakes: +5 points (total) for doing Gram-Schmidt.

26 MAT Test SOLUTIONS, FORM MAKE-UP. Let W be the subspace spanned by,,. Note that this basis is not orthogonal. 6 a. [5 points] Find the orthogonal projection of into W. c = ( A A ) ( A B ) = p = A c = Grading: + points for finding A and u, + points for the formula, + points for the calculation. Grading for common mistakes: 5 points for only finding the coordinates. b. [5 points] Find an orthogonal basis for W. NEW = OLD = NEW = OLD OLD NEW NEW = = NEW NEW NEW = OLD OLD NEW NEW OLD NEW NEW = 9 = NEW NEW NEW NEW 9 Grading: +5 points for each vector. Grading for common mistakes: +5 points (total) for dividing each vector by its length.

27 MAT Test SOLUTIONS, FORM MAKE-UP. Do the following, for the following set of data points: ( 5, 5), (, 7), (, 9), (, 9). a. [ points] Find the parabola y = ax + bx + c which best fits these points B = 9 9 a b = (A A) (A B) = 8/89 5/89 c 785/89 y = 8 89 x x = (.597) x + (8.7756) x +.5 Grading: + points for A, + points for B, + points for finding the coefficients, + points for the equation y =. b. [ points] Find the parabola y = ax + c with no linear term which best fits these points B = 9 9 a = (A A) (A B) = c 7/ 557/ y = 7 x = (.987) x Grading: + points for A, + points for B, + points for finding the coefficients, + points for the equation y =.

28 MAT Test SOLUTIONS, FORM MAKE-UP. [ points] Find the Least Squares Solution to the following system of linear equations: x + x + x = x x x = x x 5x = 5x x = ˆx = ( A A ) ( A B ) = 88/5 8/5 = / Grading: +5 points for writing down A and B, + points for the normal equation, + points for the calculation. Grading for common mistakes: points for finding Aˆx. 5. [5 points] Find a basis for W, the orthogonal complement of W, if W is the subspace spanned by, Parameterized by: A = RREF x x x = x x + x = / / x / x / = α + β x x Grading: +5 points for A, + points the null space basis. Grading for common mistakes: +5 points (total) for Gram-Schmidt.

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