MAT 242 Test 3 SOLUTIONS, FORM A
|
|
- Mary King
- 7 years ago
- Views:
Transcription
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.
MAT 242 Test 2 SOLUTIONS, FORM T
MAT 242 Test 2 SOLUTIONS, FORM T 5 3 5 3 3 3 3. Let v =, v 5 2 =, v 3 =, and v 5 4 =. 3 3 7 3 a. [ points] The set { v, v 2, v 3, v 4 } is linearly dependent. Find a nontrivial linear combination of these
More informationInner product. Definition of inner product
Math 20F Linear Algebra Lecture 25 1 Inner product Review: Definition of inner product. Slide 1 Norm and distance. Orthogonal vectors. Orthogonal complement. Orthogonal basis. Definition of inner product
More informationInner Product Spaces
Math 571 Inner Product Spaces 1. Preliminaries An inner product space is a vector space V along with a function, called an inner product which associates each pair of vectors u, v with a scalar u, v, and
More informationα = u v. In other words, Orthogonal Projection
Orthogonal Projection Given any nonzero vector v, it is possible to decompose an arbitrary vector u into a component that points in the direction of v and one that points in a direction orthogonal to v
More informationMAT 200, Midterm Exam Solution. a. (5 points) Compute the determinant of the matrix A =
MAT 200, Midterm Exam Solution. (0 points total) a. (5 points) Compute the determinant of the matrix 2 2 0 A = 0 3 0 3 0 Answer: det A = 3. The most efficient way is to develop the determinant along the
More information5. Orthogonal matrices
L Vandenberghe EE133A (Spring 2016) 5 Orthogonal matrices matrices with orthonormal columns orthogonal matrices tall matrices with orthonormal columns complex matrices with orthonormal columns 5-1 Orthonormal
More information1 2 3 1 1 2 x = + x 2 + x 4 1 0 1
(d) If the vector b is the sum of the four columns of A, write down the complete solution to Ax = b. 1 2 3 1 1 2 x = + x 2 + x 4 1 0 0 1 0 1 2. (11 points) This problem finds the curve y = C + D 2 t which
More informationProblem Set 5 Due: In class Thursday, Oct. 18 Late papers will be accepted until 1:00 PM Friday.
Math 312, Fall 2012 Jerry L. Kazdan Problem Set 5 Due: In class Thursday, Oct. 18 Late papers will be accepted until 1:00 PM Friday. In addition to the problems below, you should also know how to solve
More informationRecall that two vectors in are perpendicular or orthogonal provided that their dot
Orthogonal Complements and Projections Recall that two vectors in are perpendicular or orthogonal provided that their dot product vanishes That is, if and only if Example 1 The vectors in are orthogonal
More informationx + y + z = 1 2x + 3y + 4z = 0 5x + 6y + 7z = 3
Math 24 FINAL EXAM (2/9/9 - SOLUTIONS ( Find the general solution to the system of equations 2 4 5 6 7 ( r 2 2r r 2 r 5r r x + y + z 2x + y + 4z 5x + 6y + 7z 2 2 2 2 So x z + y 2z 2 and z is free. ( r
More informationSALEM COMMUNITY COLLEGE Carneys Point, New Jersey 08069 COURSE SYLLABUS COVER SHEET. Action Taken (Please Check One) New Course Initiated
SALEM COMMUNITY COLLEGE Carneys Point, New Jersey 08069 COURSE SYLLABUS COVER SHEET Course Title Course Number Department Linear Algebra Mathematics MAT-240 Action Taken (Please Check One) New Course Initiated
More informationNotes on Orthogonal and Symmetric Matrices MENU, Winter 2013
Notes on Orthogonal and Symmetric Matrices MENU, Winter 201 These notes summarize the main properties and uses of orthogonal and symmetric matrices. We covered quite a bit of material regarding these topics,
More informationOrthogonal Projections and Orthonormal Bases
CS 3, HANDOUT -A, 3 November 04 (adjusted on 7 November 04) Orthogonal Projections and Orthonormal Bases (continuation of Handout 07 of 6 September 04) Definition (Orthogonality, length, unit vectors).
More informationChapter 6. Orthogonality
6.3 Orthogonal Matrices 1 Chapter 6. Orthogonality 6.3 Orthogonal Matrices Definition 6.4. An n n matrix A is orthogonal if A T A = I. Note. We will see that the columns of an orthogonal matrix must be
More information17. Inner product spaces Definition 17.1. Let V be a real vector space. An inner product on V is a function
17. Inner product spaces Definition 17.1. Let V be a real vector space. An inner product on V is a function, : V V R, which is symmetric, that is u, v = v, u. bilinear, that is linear (in both factors):
More informationis in plane V. However, it may be more convenient to introduce a plane coordinate system in V.
.4 COORDINATES EXAMPLE Let V be the plane in R with equation x +2x 2 +x 0, a two-dimensional subspace of R. We can describe a vector in this plane by its spatial (D)coordinates; for example, vector x 5
More informationRecall the basic property of the transpose (for any A): v A t Aw = v w, v, w R n.
ORTHOGONAL MATRICES Informally, an orthogonal n n matrix is the n-dimensional analogue of the rotation matrices R θ in R 2. When does a linear transformation of R 3 (or R n ) deserve to be called a rotation?
More information3. INNER PRODUCT SPACES
. INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space.
More information( ) which must be a vector
MATH 37 Linear Transformations from Rn to Rm Dr. Neal, WKU Let T : R n R m be a function which maps vectors from R n to R m. Then T is called a linear transformation if the following two properties are
More informationOrthogonal Projections
Orthogonal Projections and Reflections (with exercises) by D. Klain Version.. Corrections and comments are welcome! Orthogonal Projections Let X,..., X k be a family of linearly independent (column) vectors
More informationLecture 5: Singular Value Decomposition SVD (1)
EEM3L1: Numerical and Analytical Techniques Lecture 5: Singular Value Decomposition SVD (1) EE3L1, slide 1, Version 4: 25-Sep-02 Motivation for SVD (1) SVD = Singular Value Decomposition Consider the system
More informationOrthogonal Diagonalization of Symmetric Matrices
MATH10212 Linear Algebra Brief lecture notes 57 Gram Schmidt Process enables us to find an orthogonal basis of a subspace. Let u 1,..., u k be a basis of a subspace V of R n. We begin the process of finding
More informationUniversity of Tampere Computer Graphics 2013 School of Information Sciences Exercise 2 7.2.2013
University of Tampere Computer Graphics 2013 School of Information Sciences Exercise 2 7.2.2013 1. We can get the minmum of parabola f(x) = ax 2 + bx + c,a > 0,, when f (x) = 0. In this case x has the
More informationLinear Algebra Review. Vectors
Linear Algebra Review By Tim K. Marks UCSD Borrows heavily from: Jana Kosecka kosecka@cs.gmu.edu http://cs.gmu.edu/~kosecka/cs682.html Virginia de Sa Cogsci 8F Linear Algebra review UCSD Vectors The length
More informationCS3220 Lecture Notes: QR factorization and orthogonal transformations
CS3220 Lecture Notes: QR factorization and orthogonal transformations Steve Marschner Cornell University 11 March 2009 In this lecture I ll talk about orthogonal matrices and their properties, discuss
More informationSimilarity and Diagonalization. Similar Matrices
MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that
More informationSection 6.1 - Inner Products and Norms
Section 6.1 - Inner Products and Norms Definition. Let V be a vector space over F {R, C}. An inner product on V is a function that assigns, to every ordered pair of vectors x and y in V, a scalar in F,
More informationMethods for Finding Bases
Methods for Finding Bases Bases for the subspaces of a matrix Row-reduction methods can be used to find bases. Let us now look at an example illustrating how to obtain bases for the row space, null space,
More informationMATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set.
MATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set. Vector space A vector space is a set V equipped with two operations, addition V V (x,y) x + y V and scalar
More informationLINEAR ALGEBRA. September 23, 2010
LINEAR ALGEBRA September 3, 00 Contents 0. LU-decomposition.................................... 0. Inverses and Transposes................................. 0.3 Column Spaces and NullSpaces.............................
More informationApplied Linear Algebra I Review page 1
Applied Linear Algebra Review 1 I. Determinants A. Definition of a determinant 1. Using sum a. Permutations i. Sign of a permutation ii. Cycle 2. Uniqueness of the determinant function in terms of properties
More informationLecture 5 Least-squares
EE263 Autumn 2007-08 Stephen Boyd Lecture 5 Least-squares least-squares (approximate) solution of overdetermined equations projection and orthogonality principle least-squares estimation BLUE property
More information18.06 Problem Set 4 Solution Due Wednesday, 11 March 2009 at 4 pm in 2-106. Total: 175 points.
806 Problem Set 4 Solution Due Wednesday, March 2009 at 4 pm in 2-06 Total: 75 points Problem : A is an m n matrix of rank r Suppose there are right-hand-sides b for which A x = b has no solution (a) What
More information1 VECTOR SPACES AND SUBSPACES
1 VECTOR SPACES AND SUBSPACES What is a vector? Many are familiar with the concept of a vector as: Something which has magnitude and direction. an ordered pair or triple. a description for quantities such
More information4.3 Least Squares Approximations
18 Chapter. Orthogonality.3 Least Squares Approximations It often happens that Ax D b has no solution. The usual reason is: too many equations. The matrix has more rows than columns. There are more equations
More informationAu = = = 3u. Aw = = = 2w. so the action of A on u and w is very easy to picture: it simply amounts to a stretching by 3 and 2, respectively.
Chapter 7 Eigenvalues and Eigenvectors In this last chapter of our exploration of Linear Algebra we will revisit eigenvalues and eigenvectors of matrices, concepts that were already introduced in Geometry
More informationSolutions to Math 51 First Exam January 29, 2015
Solutions to Math 5 First Exam January 29, 25. ( points) (a) Complete the following sentence: A set of vectors {v,..., v k } is defined to be linearly dependent if (2 points) there exist c,... c k R, not
More informationVector Spaces 4.4 Spanning and Independence
Vector Spaces 4.4 and Independence October 18 Goals Discuss two important basic concepts: Define linear combination of vectors. Define Span(S) of a set S of vectors. Define linear Independence of a set
More informationReview Jeopardy. Blue vs. Orange. Review Jeopardy
Review Jeopardy Blue vs. Orange Review Jeopardy Jeopardy Round Lectures 0-3 Jeopardy Round $200 How could I measure how far apart (i.e. how different) two observations, y 1 and y 2, are from each other?
More informationInner Product Spaces and Orthogonality
Inner Product Spaces and Orthogonality week 3-4 Fall 2006 Dot product of R n The inner product or dot product of R n is a function, defined by u, v a b + a 2 b 2 + + a n b n for u a, a 2,, a n T, v b,
More informationT ( a i x i ) = a i T (x i ).
Chapter 2 Defn 1. (p. 65) Let V and W be vector spaces (over F ). We call a function T : V W a linear transformation form V to W if, for all x, y V and c F, we have (a) T (x + y) = T (x) + T (y) and (b)
More informationMath 333 - Practice Exam 2 with Some Solutions
Math 333 - Practice Exam 2 with Some Solutions (Note that the exam will NOT be this long) Definitions (0 points) Let T : V W be a transformation Let A be a square matrix (a) Define T is linear (b) Define
More information4 MT210 Notebook 4 3. 4.1 Eigenvalues and Eigenvectors... 3. 4.1.1 Definitions; Graphical Illustrations... 3
MT Notebook Fall / prepared by Professor Jenny Baglivo c Copyright 9 by Jenny A. Baglivo. All Rights Reserved. Contents MT Notebook. Eigenvalues and Eigenvectors................................... Definitions;
More information1 Introduction to Matrices
1 Introduction to Matrices In this section, important definitions and results from matrix algebra that are useful in regression analysis are introduced. While all statements below regarding the columns
More informationLectures notes on orthogonal matrices (with exercises) 92.222 - Linear Algebra II - Spring 2004 by D. Klain
Lectures notes on orthogonal matrices (with exercises) 92.222 - Linear Algebra II - Spring 2004 by D. Klain 1. Orthogonal matrices and orthonormal sets An n n real-valued matrix A is said to be an orthogonal
More information[1] Diagonal factorization
8.03 LA.6: Diagonalization and Orthogonal Matrices [ Diagonal factorization [2 Solving systems of first order differential equations [3 Symmetric and Orthonormal Matrices [ Diagonal factorization Recall:
More informationLecture 14: Section 3.3
Lecture 14: Section 3.3 Shuanglin Shao October 23, 2013 Definition. Two nonzero vectors u and v in R n are said to be orthogonal (or perpendicular) if u v = 0. We will also agree that the zero vector in
More informationMath 312 Homework 1 Solutions
Math 31 Homework 1 Solutions Last modified: July 15, 01 This homework is due on Thursday, July 1th, 01 at 1:10pm Please turn it in during class, or in my mailbox in the main math office (next to 4W1) Please
More informationSection 4.4 Inner Product Spaces
Section 4.4 Inner Product Spaces In our discussion of vector spaces the specific nature of F as a field, other than the fact that it is a field, has played virtually no role. In this section we no longer
More informationSolving Quadratic Equations by Factoring
4.7 Solving Quadratic Equations by Factoring 4.7 OBJECTIVE 1. Solve quadratic equations by factoring The factoring techniques you have learned provide us with tools for solving equations that can be written
More information( ) FACTORING. x In this polynomial the only variable in common to all is x.
FACTORING Factoring is similar to breaking up a number into its multiples. For example, 10=5*. The multiples are 5 and. In a polynomial it is the same way, however, the procedure is somewhat more complicated
More informationQuestion 2: How do you solve a matrix equation using the matrix inverse?
Question : How do you solve a matrix equation using the matrix inverse? In the previous question, we wrote systems of equations as a matrix equation AX B. In this format, the matrix A contains the coefficients
More informationMatrices and Linear Algebra
Chapter 2 Matrices and Linear Algebra 2. Basics Definition 2... A matrix is an m n array of scalars from a given field F. The individual values in the matrix are called entries. Examples. 2 3 A = 2 4 2
More informationMath 550 Notes. Chapter 7. Jesse Crawford. Department of Mathematics Tarleton State University. Fall 2010
Math 550 Notes Chapter 7 Jesse Crawford Department of Mathematics Tarleton State University Fall 2010 (Tarleton State University) Math 550 Chapter 7 Fall 2010 1 / 34 Outline 1 Self-Adjoint and Normal Operators
More informationα α λ α = = λ λ α ψ = = α α α λ λ ψ α = + β = > θ θ β > β β θ θ θ β θ β γ θ β = γ θ > β > γ θ β γ = θ β = θ β = θ β = β θ = β β θ = = = β β θ = + α α α α α = = λ λ λ λ λ λ λ = λ λ α α α α λ ψ + α =
More informationThe Singular Value Decomposition in Symmetric (Löwdin) Orthogonalization and Data Compression
The Singular Value Decomposition in Symmetric (Löwdin) Orthogonalization and Data Compression The SVD is the most generally applicable of the orthogonal-diagonal-orthogonal type matrix decompositions Every
More information160 CHAPTER 4. VECTOR SPACES
160 CHAPTER 4. VECTOR SPACES 4. Rank and Nullity In this section, we look at relationships between the row space, column space, null space of a matrix and its transpose. We will derive fundamental results
More informationLinearly Independent Sets and Linearly Dependent Sets
These notes closely follow the presentation of the material given in David C. Lay s textbook Linear Algebra and its Applications (3rd edition). These notes are intended primarily for in-class presentation
More informationLinear Algebra: Vectors
A Linear Algebra: Vectors A Appendix A: LINEAR ALGEBRA: VECTORS TABLE OF CONTENTS Page A Motivation A 3 A2 Vectors A 3 A2 Notational Conventions A 4 A22 Visualization A 5 A23 Special Vectors A 5 A3 Vector
More informationThese axioms must hold for all vectors ū, v, and w in V and all scalars c and d.
DEFINITION: A vector space is a nonempty set V of objects, called vectors, on which are defined two operations, called addition and multiplication by scalars (real numbers), subject to the following axioms
More informationNumerical Methods I Solving Linear Systems: Sparse Matrices, Iterative Methods and Non-Square Systems
Numerical Methods I Solving Linear Systems: Sparse Matrices, Iterative Methods and Non-Square Systems Aleksandar Donev Courant Institute, NYU 1 donev@courant.nyu.edu 1 Course G63.2010.001 / G22.2420-001,
More informationMatrix Representations of Linear Transformations and Changes of Coordinates
Matrix Representations of Linear Transformations and Changes of Coordinates 01 Subspaces and Bases 011 Definitions A subspace V of R n is a subset of R n that contains the zero element and is closed under
More informationSpringfield Technical Community College School of Mathematics, Sciences & Engineering Transfer
Springfield Technical Community College School of Mathematics, Sciences & Engineering Transfer Department: Mathematics Course Title: Algebra 2 Course Number: MAT-097 Semester: Fall 2015 Credits: 3 Non-Graduation
More informationSection 5.3. Section 5.3. u m ] l jj. = l jj u j + + l mj u m. v j = [ u 1 u j. l mj
Section 5. l j v j = [ u u j u m ] l jj = l jj u j + + l mj u m. l mj Section 5. 5.. Not orthogonal, the column vectors fail to be perpendicular to each other. 5..2 his matrix is orthogonal. Check that
More informationExamination paper for TMA4115 Matematikk 3
Department of Mathematical Sciences Examination paper for TMA45 Matematikk 3 Academic contact during examination: Antoine Julien a, Alexander Schmeding b, Gereon Quick c Phone: a 73 59 77 82, b 40 53 99
More informationLinear Algebra Done Wrong. Sergei Treil. Department of Mathematics, Brown University
Linear Algebra Done Wrong Sergei Treil Department of Mathematics, Brown University Copyright c Sergei Treil, 2004, 2009, 2011, 2014 Preface The title of the book sounds a bit mysterious. Why should anyone
More informationFinite Dimensional Hilbert Spaces and Linear Inverse Problems
Finite Dimensional Hilbert Spaces and Linear Inverse Problems ECE 174 Lecture Supplement Spring 2009 Ken Kreutz-Delgado Electrical and Computer Engineering Jacobs School of Engineering University of California,
More informationMATH 304 Linear Algebra Lecture 18: Rank and nullity of a matrix.
MATH 304 Linear Algebra Lecture 18: Rank and nullity of a matrix. Nullspace Let A = (a ij ) be an m n matrix. Definition. The nullspace of the matrix A, denoted N(A), is the set of all n-dimensional column
More informationMATH 423 Linear Algebra II Lecture 38: Generalized eigenvectors. Jordan canonical form (continued).
MATH 423 Linear Algebra II Lecture 38: Generalized eigenvectors Jordan canonical form (continued) Jordan canonical form A Jordan block is a square matrix of the form λ 1 0 0 0 0 λ 1 0 0 0 0 λ 0 0 J = 0
More informationMathematics. (www.tiwariacademy.com : Focus on free Education) (Chapter 5) (Complex Numbers and Quadratic Equations) (Class XI)
( : Focus on free Education) Miscellaneous Exercise on chapter 5 Question 1: Evaluate: Answer 1: 1 ( : Focus on free Education) Question 2: For any two complex numbers z1 and z2, prove that Re (z1z2) =
More informationMean value theorem, Taylors Theorem, Maxima and Minima.
MA 001 Preparatory Mathematics I. Complex numbers as ordered pairs. Argand s diagram. Triangle inequality. De Moivre s Theorem. Algebra: Quadratic equations and express-ions. Permutations and Combinations.
More informationNOTES ON LINEAR TRANSFORMATIONS
NOTES ON LINEAR TRANSFORMATIONS Definition 1. Let V and W be vector spaces. A function T : V W is a linear transformation from V to W if the following two properties hold. i T v + v = T v + T v for all
More informationSubspaces of R n LECTURE 7. 1. Subspaces
LECTURE 7 Subspaces of R n Subspaces Definition 7 A subset W of R n is said to be closed under vector addition if for all u, v W, u + v is also in W If rv is in W for all vectors v W and all scalars r
More informationMATH 304 Linear Algebra Lecture 20: Inner product spaces. Orthogonal sets.
MATH 304 Linear Algebra Lecture 20: Inner product spaces. Orthogonal sets. Norm The notion of norm generalizes the notion of length of a vector in R n. Definition. Let V be a vector space. A function α
More informationEquations, Inequalities & Partial Fractions
Contents Equations, Inequalities & Partial Fractions.1 Solving Linear Equations 2.2 Solving Quadratic Equations 1. Solving Polynomial Equations 1.4 Solving Simultaneous Linear Equations 42.5 Solving Inequalities
More informationAPPLICATIONS. are symmetric, but. are not.
CHAPTER III APPLICATIONS Real Symmetric Matrices The most common matrices we meet in applications are symmetric, that is, they are square matrices which are equal to their transposes In symbols, A t =
More informationMore than you wanted to know about quadratic forms
CALIFORNIA INSTITUTE OF TECHNOLOGY Division of the Humanities and Social Sciences More than you wanted to know about quadratic forms KC Border Contents 1 Quadratic forms 1 1.1 Quadratic forms on the unit
More informationISOMETRIES OF R n KEITH CONRAD
ISOMETRIES OF R n KEITH CONRAD 1. Introduction An isometry of R n is a function h: R n R n that preserves the distance between vectors: h(v) h(w) = v w for all v and w in R n, where (x 1,..., x n ) = x
More informationMATHEMATICAL METHODS OF STATISTICS
MATHEMATICAL METHODS OF STATISTICS By HARALD CRAMER TROFESSOK IN THE UNIVERSITY OF STOCKHOLM Princeton PRINCETON UNIVERSITY PRESS 1946 TABLE OF CONTENTS. First Part. MATHEMATICAL INTRODUCTION. CHAPTERS
More informationMultidimensional data and factorial methods
Multidimensional data and factorial methods Bidimensional data x 5 4 3 4 X 3 6 X 3 5 4 3 3 3 4 5 6 x Cartesian plane Multidimensional data n X x x x n X x x x n X m x m x m x nm Factorial plane Interpretation
More informationLinear algebra and the geometry of quadratic equations. Similarity transformations and orthogonal matrices
MATH 30 Differential Equations Spring 006 Linear algebra and the geometry of quadratic equations Similarity transformations and orthogonal matrices First, some things to recall from linear algebra Two
More information1.3. DOT PRODUCT 19. 6. If θ is the angle (between 0 and π) between two non-zero vectors u and v,
1.3. DOT PRODUCT 19 1.3 Dot Product 1.3.1 Definitions and Properties The dot product is the first way to multiply two vectors. The definition we will give below may appear arbitrary. But it is not. It
More informationx y The matrix form, the vector form, and the augmented matrix form, respectively, for the system of equations are
Solving Sstems of Linear Equations in Matri Form with rref Learning Goals Determine the solution of a sstem of equations from the augmented matri Determine the reduced row echelon form of the augmented
More information3. Let A and B be two n n orthogonal matrices. Then prove that AB and BA are both orthogonal matrices. Prove a similar result for unitary matrices.
Exercise 1 1. Let A be an n n orthogonal matrix. Then prove that (a) the rows of A form an orthonormal basis of R n. (b) the columns of A form an orthonormal basis of R n. (c) for any two vectors x,y R
More informationName: Section Registered In:
Name: Section Registered In: Math 125 Exam 3 Version 1 April 24, 2006 60 total points possible 1. (5pts) Use Cramer s Rule to solve 3x + 4y = 30 x 2y = 8. Be sure to show enough detail that shows you are
More information3.1. Solving linear equations. Introduction. Prerequisites. Learning Outcomes. Learning Style
Solving linear equations 3.1 Introduction Many problems in engineering reduce to the solution of an equation or a set of equations. An equation is a type of mathematical expression which contains one or
More information2.3 Convex Constrained Optimization Problems
42 CHAPTER 2. FUNDAMENTAL CONCEPTS IN CONVEX OPTIMIZATION Theorem 15 Let f : R n R and h : R R. Consider g(x) = h(f(x)) for all x R n. The function g is convex if either of the following two conditions
More informationThe Geometry of Polynomial Division and Elimination
The Geometry of Polynomial Division and Elimination Kim Batselier, Philippe Dreesen Bart De Moor Katholieke Universiteit Leuven Department of Electrical Engineering ESAT/SCD/SISTA/SMC May 2012 1 / 26 Outline
More informationMATH 551 - APPLIED MATRIX THEORY
MATH 55 - APPLIED MATRIX THEORY FINAL TEST: SAMPLE with SOLUTIONS (25 points NAME: PROBLEM (3 points A web of 5 pages is described by a directed graph whose matrix is given by A Do the following ( points
More informationAlgebra 2: Q1 & Q2 Review
Name: Class: Date: ID: A Algebra 2: Q1 & Q2 Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Which is the graph of y = 2(x 2) 2 4? a. c. b. d. Short
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +
More informationCITY UNIVERSITY LONDON. BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION
No: CITY UNIVERSITY LONDON BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION ENGINEERING MATHEMATICS 2 (resit) EX2005 Date: August
More informationby the matrix A results in a vector which is a reflection of the given
Eigenvalues & Eigenvectors Example Suppose Then So, geometrically, multiplying a vector in by the matrix A results in a vector which is a reflection of the given vector about the y-axis We observe that
More informationLinear Algebraic Equations, SVD, and the Pseudo-Inverse
Linear Algebraic Equations, SVD, and the Pseudo-Inverse Philip N. Sabes October, 21 1 A Little Background 1.1 Singular values and matrix inversion For non-smmetric matrices, the eigenvalues and singular
More informationLecture 2 Matrix Operations
Lecture 2 Matrix Operations transpose, sum & difference, scalar multiplication matrix multiplication, matrix-vector product matrix inverse 2 1 Matrix transpose transpose of m n matrix A, denoted A T or
More informationJUST THE MATHS UNIT NUMBER 1.8. ALGEBRA 8 (Polynomials) A.J.Hobson
JUST THE MATHS UNIT NUMBER 1.8 ALGEBRA 8 (Polynomials) by A.J.Hobson 1.8.1 The factor theorem 1.8.2 Application to quadratic and cubic expressions 1.8.3 Cubic equations 1.8.4 Long division of polynomials
More informationAlgebraic Concepts Algebraic Concepts Writing
Curriculum Guide: Algebra 2/Trig (AR) 2 nd Quarter 8/7/2013 2 nd Quarter, Grade 9-12 GRADE 9-12 Unit of Study: Matrices Resources: Textbook: Algebra 2 (Holt, Rinehart & Winston), Ch. 4 Length of Study:
More informationLinear Maps. Isaiah Lankham, Bruno Nachtergaele, Anne Schilling (February 5, 2007)
MAT067 University of California, Davis Winter 2007 Linear Maps Isaiah Lankham, Bruno Nachtergaele, Anne Schilling (February 5, 2007) As we have discussed in the lecture on What is Linear Algebra? one of
More informationThe Determinant: a Means to Calculate Volume
The Determinant: a Means to Calculate Volume Bo Peng August 20, 2007 Abstract This paper gives a definition of the determinant and lists many of its well-known properties Volumes of parallelepipeds are
More informationPart 2: Analysis of Relationship Between Two Variables
Part 2: Analysis of Relationship Between Two Variables Linear Regression Linear correlation Significance Tests Multiple regression Linear Regression Y = a X + b Dependent Variable Independent Variable
More information