Chapter 6: Orthogonality


 Heather Ellis
 1 years ago
 Views:
Transcription
1 Chapter 6: Orthogonality (Last Updated: November 7, 7) These notes are derived primarily from Linear Algebra and its applications by David Lay (4ed). A few theorems have been moved around.. Inner products We now return to a discussion on the geometry of vectors. There are many applications of the notion of orthogonality, some of which we will discuss. A basic (geometric) question that we will address shortly is the following. Suppose you are given a plane P and a point p (in R 3 ). What is the distance from p to P? That is, what is the length of the shortest possible line segment that one could draw from p to P. Definition. Let u, v R n. The inner product of u and v is defined as u v = u T v = [u u n. = u v + + u n v n. The inner product is also referred to as the dot product. Another product, the cross product, will be discussed at a later time. Example. Let a = 3 5 and b = Theorem. Let u, v, w R n and let c R. Then () u v = v u. () (u + v) w = (u w) + (v w). v v n. Compute a b and b a. (3) (cu) v = c(u v). The inner product is a useful tool to study the geometry of vectors. (4) u u and u u = if and only if u =. Definition. The length (or norm) of v R n is the nonnegative scalar v defined by v = v v = v + + v n and v = v v. A unit vector is a vector of length. Note that cv = c v for c R. If v, then u = v v is the unit vector in the same direction as v. Example 3. Find the lengths of a and b in Example and find their associated unit vectors.
2 Recall that the distance between two points (a, b ) and (a, b ) in R is determined by the wellknown distance formula (a a ) + (b b ). We can similarly define distance between vectors. Definition 3. For u, v R n, the distance between u and v, written d(u, v), is the length of the vector u v. That is, d(u, v) = u v. Example 4. Find the distance between a and b in Example. Definition 4. Two vectors u, v R n are said to be orthogonal (to each other) if u v =. Orthogonality generalizes the idea of perpendicular lines in R. Two lines (represented as vectors u, v) are perpendicular if and only if the distance from u to v equals the distance from u to v. (d(u, v)) = u ( v) = u + v = (u + v) (u + v) = u + u v + v (d(u, v)) = u v = (u v) (u v) = u u v + v. Hence, these two quantities are equal if and only if u v = u v. Equivalently, u v =. The next theorem now follows directly. Theorem 5 (The Pythagorean Theorem). Two vectors u, v R n are orthogonal if and only if u + v = u + v. Definition 5. Let W R n be a subspace. The set W = {z R n : z w = for all w W } is called the orthogonal complement of W. Part of your homework will be to show that W is a subspace of R n. Example 6. Let W be a plane through the origin in R 3 and let L be a line through perpendicular to W. If z L and w W are nonzero, then the line segment from to z is perpendicular to the line segment from to w. In fact, L = W and W = L. Theorem 7. Let A be an n n matrix. Then (RowA) = NulA and (ColA) = NulA T. Proof. If x NulA, then Ax = by definition. Hence, x is perpendicular to each row of A. Since the rows of A span RowA, then x (RowA). Conversely, if x (RowA), then x is orthogonal to each row of A and Ax =, so x NulA. The proof of the second statement is similar. Let u, v R be nonzero. By the Law of Cosines, u v = u + v u v cos θ. Rearranging gives u v cos θ = [ u + v u v = [ (u + u ) + (v + v) (u v ) (u v ) = u v + u v = u v. Hence, cos θ = u v u v
3 . Orthogonal Sets Definition 6. A set of vectors {u,..., u p } in R n is said to be an orthogonal if u i u j = for all i j. If, in addition, each u i is a unit vector, then the set is said to be orthonormal. Example 8. Show that the following set is orthogonal. Is it orthonormal? If not, find a set of orthonormal vectors with the same span. 3 3, 3 3, Theorem 9. If S = {u,..., u p } is an orthogonal set of nonzero vectors in R n, then S is linearly independent and hence a basis for a subspace spanned by S. Proof. Write = c u + + c p u p. Then = u = (c u + + c p u p ) u = c (u u ) + + c p (u p u ) = c (u u ). Since u u (because u ), then c =. Repeating this argument with u,..., u p gives c = = c p =. Hence, S is linearly independent. Let {u,..., u p } be an orthogonal basis for a subspace W of R n. Let u W and write y = c u + + c p u p. Then y u i = c i (u i u i ), and so c i = y u i u i u i, i =,..., p. Example. Show that the set S below is an orthogonal basis of R 3 and express the given vector x as a linear combination of these vectors. 8 S =, 4,, x = 4. 3 Here is an easier version of the problem hinted at in the beginning of this chapter. Given a point p and a line L (in R ), what is the distance from p to L. The solution of this uses orthogonal projections. Let L = Span{u} and p given by the vector y. We need to know the length of the vector orthogonal to u through y. By translation, this is equivalent to the length of the vector z = y ŷ where ŷ = αu for some scalar α. Then = z u = (y αu) u = y u (αu) u = y u α(u u). Hence, α = y u y u u u and so ŷ = u uu. Note that if we replace u by cu for any scalar c this definition does not change and thus we have defined the projection for all of L.
4 Definition 7. Given vectors y, u R n, and L = Span{u}, the orthogonal projection of y onto L is defined as ŷ = proj L y = y u u u u. Note that this gives a decomposition of the vector y as y = ŷ + z where ŷ L and z L. Hence, every vector in R n can be written (uniquely) as the sum of an element in L and an element in L. Since L L = {}, then it follows that dim L = n. In the next section we will generalize this to larger subspaces. [ [ 4 Example. Compute the orthogonal projection of onto the line L through and the 7 origin. Use this to find the distance from y to L. Definition 8. If W is a subspace of R n spanned by an orthonormal set S = {u,..., u p }, then we say S is an orthonormal basis of W. Example. The standard basis is an orthonormal basis of R n. Theorem 3. An m n matrix U has orthonormal columns if and only if U T U = I. Proof. Write U = [u u n. Then u T u T u u T u u T u n U T u U = [u T u n = u u T u u T u n.. u T.... n u T n u u T n u u T n u n Hence, U T U = I if and only if u i u i = for all i and u i u j = for all i j. Theorem 4. Let U be an m n matrix with orthonormal columns and let x, y R n. Then () Ux = x () (Ux) (Ux) = x y (3) (Ux) (Ux) = if and only if x y =. Proof. We will prove (). The rest are left as an exercise. Write U = [u u n. Then Ux = Ux Ux = (u x + u n x n ) (u x + u n x n ) = i,j (u i x i ) (u j x j ) = i,j x i x j (u i u j ) = i x i (u i u i ) = i x i = x.
5 3. Orthogonal Projections The next definition generalizes projections onto lines. Definition 9. Let W be a subspace of R n with orthogonal basis {u,..., u p }. For y R n, the orthogonal projection of y onto W is given by proj W y = y u u u u + + y u n u n u n u n. This definition matches our previous one when W is dimensional. Note that proj W y W because it is a linear combination of basis elements. Also note that the definition simplifies when the basis {u,..., u p } is orthonormal. In this case, if we let U = [u u p, then proj W y = UU T y for all y R n. Theorem 5 (Orthogonal Decomposition Theorem). Let W be a subspace of R n with orthogonal basis {u,..., u p }. Then each y R n can be written uniquely in the form y = ŷ + z where ŷ W and z W. In fact ŷ = proj W y and z = y ŷ. Proof. Note that if W = {}, then this theorem is trivial. As noted above, proj W y W. We claim z = y ŷ W. ( ) y u z u = (y ŷ) u = y u ŷ u = y u u = y u y u =. u u It is clear that this holds similarly for u,, u p. By linearity, z y =, so z W. To prove uniqueness, let y = w + x be another decomposition with w W and x W. Then w+x = y = ŷ+z, so (w ŷ) = (z x). But (w ŷ) W and (z x) W. Since W W = {}, then w ŷ = so w = ŷ. Similarly, z = x. We will show in the next section that every subspace has an orthogonal basis. Corollary 6. Let W be a subspace of R n with orthogonal basis {u,..., u p }. Then y W if and only if proj W y = y. Example 7. Let W = Span{u, u } below. Note that u and u are orthogonal. Write y (below) as a vector ŷ W and z W. 3 u =, u =, y =. Theorem 8 (Best Approximation Theorem). Let W be a subspace of R n and y R n. Then ŷ = proj W y is the closest point to W in the sense that y ŷ < y v for all v W, v y. 6
6 4. The GramSchmidt Process Orthogonal projections give us a way to find an orthogonal basis for any W of R n. Example 9. Let W = Span{x, x } with x, x below. Construct an orthogonal basis for W. x = 3, x = 8 5. Let v = x and W = Span{v }. It suffices to find a vector v W orthogonal to W. Let p = proj W x W. Then x = p + (x p) where x p W. 6 v = x p = x x 8 9 v v = 5 v v 6 3 = 5. Now v v = and v, v W. Hence, {v, v } is a basis for W. Note that if we wanted an orthonormal basis for W then we can just take the unit vectors associated to v and v. 3 This process could continue. Say W was threedimensional. We could then let W = Span{v, v } and find the projection of x 3 onto W. We ll prove the next theorem using this idea. Theorem (The GramSchmidt Process). Given a basis {x,..., x p } for a nonzero subspace W R n, define v = x v = x x v v v v v 3 = x 3 x 3 v v v v x 3 v v v v. v p = x p x p v v x p v v x p v p v p v v v v v p v p Then {v,..., v p } is an orthogonal basis for W. In addition, Span{v,..., v k } = Span{x,..., x k } for all k p. Proof. For k p, set W k = Span{x,..., x k } and V k = Span{v,..., v k }. Since v = x. Then it (trivially) holds that W = V and {v } is orthogonal. Suppose for some k, k < n, that W k = V k and that {v,..., v k } is an orthogonal set. Define v k+ = x k+ proj Wk x k+ W k W k+.
7 By the Orthogonal Decomposition Theorem, v k+ is orthogonal to W k. Since x k+ W k+, then v k+ W k+. Hence, {v,..., v k+ } is an orthogonal set of k + nonzero vectors in W k+ and hence a basis of W k+. Hence, W k+ = V k+. The result now follows by induction. Example. Let W = Span{x, x, x 3 } with x i below. Construct an orthogonal basis for W. x =, x =, x 3 =. Set v = x. Then v = x x v v v v = 3 = /3 /3 /3. Now, v 3 = x 3 x 3 v v v v x 3 v v v v = 3 5 /3 /3 /3 = / /. Hence, an orthogonal basis for W is {v, v, v 3 }.
8 5. Leastsquares problems In data science, one often wants to be able to approximate a set of data by a curve. Possibly, one might hope to construct the line that best fits the data. This is known (by one name) as linear regression. In this section we ll study the linear algebra approach to this problem. Suppose the system Ax = b is inconsistent. Previously, we gave up all hope then of solving this system because no solution existed. However, if we give up the idea that we must find an exact solution and instead focus on finding an approximate solution, then we may have hope of solving. Definition. If A is an m n matrix and b R n, a leastsquares solution of Ax = b is a vector ˆx R n such that for all x R n, b Aˆx b Ax. Geometrically, we think of Aˆx as the projection of b onto ColA. That is, if ˆb = proj ColA b, then the equation Ax = ˆb is consistent. Let ˆx R n be a solution (there may be several). By the Best Approximation Theorem, ˆb is the point on ColA closest to b and so Aˆx is a leastsquares solution to Ax = b By the Orthogonal Decomposition Theorem, b ˆb is orthogonal to ColA. Hence, if a j is any column of A, then a j (b ˆb) =. That is, a T j (b ˆb) =. But a T j is a row of A T and so A T (b ˆb). Replacing ˆb with Aˆx and expanding we get A T Aˆx = A T b. The equations corresponding to this system are the normal equations for Ax = b. We have now essentially proven the following theorem. Theorem. The set of leastsquares solutions of Ax = b coincides with the nonempty set of solutions of the normal equations A T Ax = A T b. Example 3. Find a leastsquares solution of the inconsistent system Ax = b where 5 A = 4 and b =. 3 We will use normal equations. First we compute [ [ A T A =, A T b =. 9 5 To solve the equation A T Ax = A T b we invert A T A. [ [ [ ˆx = (A T A) A T b = 9 5 /9 = /9 Hence, when A T A is invertible then the leastsquares solution ˆx is unique and ˆx = (A T A) A T b.
9 As an application of this, we ll see how to fit data to a line using leastsquares. To match notation commonly used in statistical analysis, we denote the equation Ax = b by Xβ = y. The matrix X is referred to as the design matrix, β as the parameter vector, and y as the observation vector. Suppose we have a a set of data points (x, y ), (x, y ),..., (x n, y n ), perhaps from some experiments. We would like to model this data be a line to predict outcomes that did not appear in our experiment. Say this line is written y = β + β x. The residual of a point (x i, y i ) is the distance from that point to the line. The leastsquares line is the line that minimizes the sum of the squares of the residuals. Suppose the data was all on the line. Then they would all satisfy, β + β x = y β + β x = y. β + β x n = y n. We could write this system as Xβ = y where x x X =, β =.. x n [ β β y y, y =.. If the data does not lie on the line (and this is likely) then we want the vector β to be the leastsquares solution of Xβ = y that minimizes the distance between Xβ and y. Example 4. Find the equation y = β +β x of the leastsquares line that best fits the data points (4, ), (, ), (3, 3), (5, 5). We build the matrix X and vector y from the data, 4 X = 3, y = For the leastsquares solution of Xβ = y, we have the normal equation X T Xβ = X T y where [ [ 4 X T X =, X T y = Hence, [ β β y n [ 4/35 = (X T X) X T y =. 7/35
10 7. Diagonalization of Symmetric Matrices We have seen already that it is quite time intensive to determine whether a matrix is diagonalizable. We ll see that there are certain cases when a matrix is always diagonalizable. Definition. A matrix A is symmetric if A T = A. 3 4 Example 5. Let A = Note that A T = A, so A is symmetric. The characteristic polynomial of A is χ A (t) = (t + )(t 7) so the eigenvalues are and 7. The corresponding eigenspaces have bases, λ =,, λ = 7,,. Hence, A is diagonalizable. Now we use GramSchmidt to find an orthogonal basis for R 3. Note that the eigenvector for λ = is already orthogonal to both eigenvectors for λ = 7. / v =, v =, v 3 =. Finally, we normalize each vector, / u =, u = / /3 /3, u 3 = / /3 Now the matrix U = [u u u 3 is orthogonal and so U T U = I. /3 /3. Theorem 6. If A is symmetric, then any two eigenvectors from different eigenspaces are orthogonal. Proof. Let v, v be eigenvectors for A with corresponding eigenvalues λ, λ, λ λ. Then λ (v v ) = (λ v ) T v = (Av ) T v = v T A T v = v T Av = v T (λ v ) = λ (v v ). /3 Hence, (λ λ )(v v ) =. Since λ λ, then we must have v v =. Based on the previous theorem, we say that the eigenspaces of A are mutually orthogonal. Definition. An n n matrix A is orthogonally diagonalizable if there exists an orthogonal n n matrix P and a diagonal matrix D such that A = P DP T. Theorem 7. If A is orthogonally diagonalizable, then A is symmetric.
11 Proof. Since A is orthogonally diagonalizable, then A = P DP T for some orthogonal matrix P and diagonal matrix D. A is symmetric because A T = (P DP T ) T = (P T ) T D T P T = P DP T = A. It turns out the converse of the above theorem is also true! The set of eigenvalues of a matrix A is called the spectrum of A and is denoted σ A. Theorem 8 (The Spectral Theorem for symmetric matrices). Let A be a (real) n n symmetric matrix. Then the following hold. () A has n real eigenvalues, counting multiplicities. () For each eigenvalue λ of A, geomult λ (A) = algmult λ (A). (3) The eigenspaces are mutually orthogonal. (4) A is orthogonally diagonalizable. Proof. Every eigenvalue of a symmetric matrix is real. The second part of () as well as () are immediate consequences of (4). We proved (3) in Theorem 6. Note that (4) is trivial when A has n distinct eigenvalues by (3). We prove (4) by induction. Clearly the result holds when A is. Assume (n ) (n ) symmetric matrices are orthogonally diagonalizable. Let A be n n and let λ be an eigenvalue of A and u a (unit) eigenvector for λ. By the Gram Schmidt process, we may extend u to an orthonormal basis {u,..., u n } for R n where {u,..., u n } is a basis for W. Set U = [u u u n. Then u T U T Au u T Au n [ AU =..... = λ. B u T n Au u T n Au n The first column is as indicated because u T i Au = u T i (λu ) = λ(u i u ) = λδ ij. As U T AU is symmetric, = and B is a symmetric (n ) (n ) matrix that is orthogonally diagonalizable with eigenvalues λ,..., λ n (by the inductive hypothesis). Because A and U T AU are similar, then the eigenvalues of A are λ,..., λ n. Since B is orthogonally diagonalizable, there exists an orthogonal matrix Q such that Q T BQ = D, where the diagonal entries of D are λ,..., λ n. Now [ T [ [ [ [ λ λ λ = =. Q B Q Q T BQ D This is one of the problems on the extra credit homework assignment.
12 [ [ Note that is orthogonal. Set V = U. As the product of orthogonal matrices is Q Q orthogonal, V is itself orthogonal and V T AV is diagonal. Suppose A is orthogonally diagonalizable, so A = UDU T where U = [u u n and D is the diagonal matrix whose diagonal entries are the eigenvalues of A, λ,..., λ n. Then A = UDU T = λ u u T + + λ n u n u T n. This is known as the spectral decomposition of A. Each u i u T i (u i u T i )x is the projection of x onto Span{u i}. is called a projection matrix because Example 9. Construct a spectral decomposition of the matrix A in Example Recall that A = 6 and our orthonormal basis of Col(A) was 4 3 / u =, u = /3 /3, u 3 = /3 /3. / /3 /3 Setting U = [u u u 3 gives U T AU = D = diag(, 7, 7). The projection matrices are / / u u T =, u u T = / / The spectral decomposition is /8 /9 /8 /9 8 9 /9 /8 /9 /8, u 3u T 3 = 4/9 /9 4/9 /9 /9 /9 4/9 /9 4/9. 7u u T + 7u u T u 3 u T 3 = A.
Chapter 7: Symmetric Matrices and Quadratic Forms
Chapter 7: Symmetric Matrices and Quadratic Forms (Last Updated: December, 06) These notes are derived primarily from Linear Algebra and its applications by David Lay (4ed). A few theorems have been moved
More informationSolutions to Review Problems for Chapter 6 ( ), 7.1
Solutions to Review Problems for Chapter (, 7 The Final Exam is on Thursday, June,, : AM : AM at NESBITT Final Exam Breakdown Sections % ,79,  % 9,,7,,7  % , 7  % Let u u and v Let x x x x,
More informationWorksheet for Lecture 25 Section 6.4 GramSchmidt Process
Worksheet for Lecture Name: Section.4 GramSchmidt Process Goal For a subspace W = Span{v,..., v n }, we want to find an orthonormal basis of W. Example Let W = Span{x, x } with x = and x =. Give an orthogonal
More informationMATH 20F: LINEAR ALGEBRA LECTURE B00 (T. KEMP)
MATH 20F: LINEAR ALGEBRA LECTURE B00 (T KEMP) Definition 01 If T (x) = Ax is a linear transformation from R n to R m then Nul (T ) = {x R n : T (x) = 0} = Nul (A) Ran (T ) = {Ax R m : x R n } = {b R m
More informationMarch 27 Math 3260 sec. 56 Spring 2018
March 27 Math 3260 sec. 56 Spring 2018 Section 4.6: Rank Definition: The row space, denoted Row A, of an m n matrix A is the subspace of R n spanned by the rows of A. We now have three vector spaces associated
More informationFinal Review Written by Victoria Kala SH 6432u Office Hours R 12:30 1:30pm Last Updated 11/30/2015
Final Review Written by Victoria Kala vtkala@mathucsbedu SH 6432u Office Hours R 12:30 1:30pm Last Updated 11/30/2015 Summary This review contains notes on sections 44 47, 51 53, 61, 62, 65 For your final,
More informationLINEAR ALGEBRA SUMMARY SHEET.
LINEAR ALGEBRA SUMMARY SHEET RADON ROSBOROUGH https://intuitiveexplanationscom/linearalgebrasummarysheet/ This document is a concise collection of many of the important theorems of linear algebra, organized
More informationSection 6.2, 6.3 Orthogonal Sets, Orthogonal Projections
Section 6. 6. Orthogonal Sets Orthogonal Projections Main Ideas in these sections: Orthogonal set = A set of mutually orthogonal vectors. OG LI. Orthogonal Projection of y onto u or onto an OG set {u u
More informationMTH 2310, FALL Introduction
MTH 2310, FALL 2011 SECTION 6.2: ORTHOGONAL SETS Homework Problems: 1, 5, 9, 13, 17, 21, 23 1, 27, 29, 35 1. Introduction We have discussed previously the benefits of having a set of vectors that is linearly
More informationLeast squares problems Linear Algebra with Computer Science Application
Linear Algebra with Computer Science Application April 8, 018 1 Least Squares Problems 11 Least Squares Problems What do you do when Ax = b has no solution? Inconsistent systems arise often in applications
More informationMTH 2032 SemesterII
MTH 202 SemesterII 201011 Linear Algebra Worked Examples Dr. Tony Yee Department of Mathematics and Information Technology The Hong Kong Institute of Education December 28, 2011 ii Contents Table of Contents
More informationLINEAR ALGEBRA 1, 2012I PARTIAL EXAM 3 SOLUTIONS TO PRACTICE PROBLEMS
LINEAR ALGEBRA, I PARTIAL EXAM SOLUTIONS TO PRACTICE PROBLEMS Problem (a) For each of the two matrices below, (i) determine whether it is diagonalizable, (ii) determine whether it is orthogonally diagonalizable,
More informationDSGA 1002 Lecture notes 0 Fall Linear Algebra. These notes provide a review of basic concepts in linear algebra.
DSGA 1002 Lecture notes 0 Fall 2016 Linear Algebra These notes provide a review of basic concepts in linear algebra. 1 Vector spaces You are no doubt familiar with vectors in R 2 or R 3, i.e. [ ] 1.1
More informationOrthogonality and Least Squares
6 Orthogonality and Least Squares 6.1 INNER PRODUCT, LENGTH, AND ORTHOGONALITY INNER PRODUCT If u and v are vectors in, then we regard u and v as matrices. n 1 n The transpose u T is a 1 n matrix, and
More information6. Orthogonality and LeastSquares
Linear Algebra 6. Orthogonality and LeastSquares CSIE NCU 1 6. Orthogonality and LeastSquares 6.1 Inner product, length, and orthogonality. 2 6.2 Orthogonal sets... 8 6.3 Orthogonal projections... 13
More informationMath 4A Notes. Written by Victoria Kala Last updated June 11, 2017
Math 4A Notes Written by Victoria Kala vtkala@math.ucsb.edu Last updated June 11, 2017 Systems of Linear Equations A linear equation is an equation that can be written in the form a 1 x 1 + a 2 x 2 +...
More informationMath 3191 Applied Linear Algebra
Math 9 Applied Linear Algebra Lecture : Orthogonal Projections, GramSchmidt Stephen Billups University of Colorado at Denver Math 9Applied Linear Algebra p./ Orthonormal Sets A set of vectors {u, u,...,
More informationMath Linear Algebra
Math 220  Linear Algebra (Summer 208) Solutions to Homework #7 Exercise 6..20 (a) TRUE. u v v u = 0 is equivalent to u v = v u. The latter identity is true due to the commutative property of the inner
More informationIMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET
IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET This is a (not quite comprehensive) list of definitions and theorems given in Math 1553. Pay particular attention to the ones in red. Study Tip For each
More informationMATH 1120 (LINEAR ALGEBRA 1), FINAL EXAM FALL 2011 SOLUTIONS TO PRACTICE VERSION
MATH (LINEAR ALGEBRA ) FINAL EXAM FALL SOLUTIONS TO PRACTICE VERSION Problem (a) For each matrix below (i) find a basis for its column space (ii) find a basis for its row space (iii) determine whether
More informationWI1403LR Linear Algebra. Delft University of Technology
WI1403LR Linear Algebra Delft University of Technology Year 2013 2014 Michele Facchinelli Version 10 Last modified on February 1, 2017 Preface This summary was written for the course WI1403LR Linear
More informationMath 3191 Applied Linear Algebra
Math 191 Applied Linear Algebra Lecture 1: Inner Products, Length, Orthogonality Stephen Billups University of Colorado at Denver Math 191Applied Linear Algebra p.1/ Motivation Not all linear systems have
More informationAnnouncements Monday, November 26
Announcements Monday, November 26 Please fill out your CIOS survey! WeBWorK 6.6, 7.1, 7.2 are due on Wednesday. No quiz on Friday! But this is the only recitation on chapter 7. My office is Skiles 244
More informationReview problems for MA 54, Fall 2004.
Review problems for MA 54, Fall 2004. Below are the review problems for the final. They are mostly homework problems, or very similar. If you are comfortable doing these problems, you should be fine on
More informationIMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET
IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET This is a (not quite comprehensive) list of definitions and theorems given in Math 1553. Pay particular attention to the ones in red. Study Tip For each
More informationOrthonormal Bases; GramSchmidt Process; QRDecomposition
Orthonormal Bases; GramSchmidt Process; QRDecomposition MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 205 Motivation When working with an inner product space, the most
More informationOrthogonal Complements
Orthogonal Complements Definition Let W be a subspace of R n. If a vector z is orthogonal to every vector in W, then z is said to be orthogonal to W. The set of all such vectors z is called the orthogonal
More information6.1. Inner Product, Length and Orthogonality
These are brief notes for the lecture on Friday November 13, and Monday November 1, 2009: they are not complete, but they are a guide to what I want to say on those days. They are guaranteed to be incorrect..1.
More informationMATH 304 Linear Algebra Lecture 20: The GramSchmidt process (continued). Eigenvalues and eigenvectors.
MATH 304 Linear Algebra Lecture 20: The GramSchmidt process (continued). Eigenvalues and eigenvectors. Orthogonal sets Let V be a vector space with an inner product. Definition. Nonzero vectors v 1,v
More information(v, w) = arccos( < v, w >
MA322 Sathaye Notes on Inner Products Notes on Chapter 6 Inner product. Given a real vector space V, an inner product is defined to be a bilinear map F : V V R such that the following holds: For all v
More informationWorksheet for Lecture 23 (due December 4) Section 6.1 Inner product, length, and orthogonality
Worksheet for Lecture (due December 4) Name: Section 6 Inner product, length, and orthogonality u Definition Let u = u n product or dot product to be and v = v v n be vectors in R n We define their inner
More informationMath Linear Algebra II. 1. Inner Products and Norms
Math 342  Linear Algebra II Notes 1. Inner Products and Norms One knows from a basic introduction to vectors in R n Math 254 at OSU) that the length of a vector x = x 1 x 2... x n ) T R n, denoted x,
More informationLinear Algebra Final Exam Review
Linear Algebra Final Exam Review. Let A be invertible. Show that, if v, v, v 3 are linearly independent vectors, so are Av, Av, Av 3. NOTE: It should be clear from your answer that you know the definition.
More information22m:033 Notes: 7.1 Diagonalization of Symmetric Matrices
m:33 Notes: 7. Diagonalization of Symmetric Matrices Dennis Roseman University of Iowa Iowa City, IA http://www.math.uiowa.edu/ roseman May 3, Symmetric matrices Definition. A symmetric matrix is a matrix
More informationBASIC ALGORITHMS IN LINEAR ALGEBRA. Matrices and Applications of Gaussian Elimination. A 2 x. A T m x. A 1 x A T 1. A m x
BASIC ALGORITHMS IN LINEAR ALGEBRA STEVEN DALE CUTKOSKY Matrices and Applications of Gaussian Elimination Systems of Equations Suppose that A is an n n matrix with coefficents in a field F, and x = (x,,
More informationMATH 304 Linear Algebra Lecture 34: Review for Test 2.
MATH 304 Linear Algebra Lecture 34: Review for Test 2. Topics for Test 2 Linear transformations (Leon 4.1 4.3) Matrix transformations Matrix of a linear mapping Similar matrices Orthogonality (Leon 5.1
More informationStudy Guide for Linear Algebra Exam 2
Study Guide for Linear Algebra Exam 2 Term Vector Space Definition A Vector Space is a nonempty set V of objects, on which are defined two operations, called addition and multiplication by scalars (real
More informationTypical Problem: Compute.
Math 2040 Chapter 6 Orhtogonality and Least Squares 6.1 and some of 6.7: Inner Product, Length and Orthogonality. Definition: If x, y R n, then x y = x 1 y 1 +... + x n y n is the dot product of x and
More informationLinear Systems. Class 27. c 2008 Ron Buckmire. TITLE Projection Matrices and Orthogonal Diagonalization CURRENT READING Poole 5.4
Linear Systems Math Spring 8 c 8 Ron Buckmire Fowler 9 MWF 9: am  :5 am http://faculty.oxy.edu/ron/math//8/ Class 7 TITLE Projection Matrices and Orthogonal Diagonalization CURRENT READING Poole 5. Summary
More informationMATH 31  ADDITIONAL PRACTICE PROBLEMS FOR FINAL
MATH 3  ADDITIONAL PRACTICE PROBLEMS FOR FINAL MAIN TOPICS FOR THE FINAL EXAM:. Vectors. Dot product. Cross product. Geometric applications. 2. Row reduction. Null space, column space, row space, left
More informationApplied Linear Algebra in Geoscience Using MATLAB
Applied Linear Algebra in Geoscience Using MATLAB Contents Getting Started Creating Arrays Mathematical Operations with Arrays Using Script Files and Managing Data TwoDimensional Plots Programming in
More informationAnnouncements Monday, November 20
Announcements Monday, November 20 You already have your midterms! Course grades will be curved at the end of the semester. The percentage of A s, B s, and C s to be awarded depends on many factors, and
More informationChapter 4 Euclid Space
Chapter 4 Euclid Space Inner Product Spaces Definition.. Let V be a real vector space over IR. A real inner product on V is a real valued function on V V, denoted by (, ), which satisfies () (x, y) = (y,
More informationMATH 115A: SAMPLE FINAL SOLUTIONS
MATH A: SAMPLE FINAL SOLUTIONS JOE HUGHES. Let V be the set of all functions f : R R such that f( x) = f(x) for all x R. Show that V is a vector space over R under the usual addition and scalar multiplication
More information. = V c = V [x]v (5.1) c 1. c k
Chapter 5 Linear Algebra It can be argued that all of linear algebra can be understood using the four fundamental subspaces associated with a matrix Because they form the foundation on which we later work,
More informationPRACTICE PROBLEMS FOR THE FINAL
PRACTICE PROBLEMS FOR THE FINAL Here are a slew of practice problems for the final culled from old exams:. Let P be the vector space of polynomials of degree at most. Let B = {, (t ), t + t }. (a) Show
More informationHomework 11 Solutions. Math 110, Fall 2013.
Homework 11 Solutions Math 110, Fall 2013 1 a) Suppose that T were selfadjoint Then, the Spectral Theorem tells us that there would exist an orthonormal basis of P 2 (R), (p 1, p 2, p 3 ), consisting
More informationMath 102, Winter Final Exam Review. Chapter 1. Matrices and Gaussian Elimination
Math 0, Winter 07 Final Exam Review Chapter. Matrices and Gaussian Elimination { x + x =,. Different forms of a system of linear equations. Example: The x + 4x = 4. [ ] [ ] [ ] vector form (or the column
More information1 Last time: leastsquares problems
MATH Linear algebra (Fall 07) Lecture Last time: leastsquares problems Definition. If A is an m n matrix and b R m, then a leastsquares solution to the linear system Ax = b is a vector x R n such that
More informationMATH 1553 SAMPLE FINAL EXAM, SPRING 2018
MATH 1553 SAMPLE FINAL EXAM, SPRING 2018 Name Circle the name of your instructor below: Fathi Jankowski Kordek Strenner Yan Please read all instructions carefully before beginning Each problem is worth
More informationDiagonalizing Matrices
Diagonalizing Matrices Massoud Malek A A Let A = A k be an n n nonsingular matrix and let B = A = [B, B,, B k,, B n ] Then A n A B = A A 0 0 A k [B, B,, B k,, B n ] = 0 0 = I n 0 A n Notice that A i B
More informationLINEAR ALGEBRA W W L CHEN
LINEAR ALGEBRA W W L CHEN c W W L Chen, 1997, 2008. This chapter is available free to all individuals, on the understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied,
More informationORTHOGONALITY AND LEASTSQUARES [CHAP. 6]
ORTHOGONALITY AND LEASTSQUARES [CHAP. 6] Inner products and Norms Inner product or dot product of 2 vectors u and v in R n : u.v = u 1 v 1 + u 2 v 2 + + u n v n Calculate u.v when u = 1 2 2 0 v = 1 0
More informationLINEAR ALGEBRA BOOT CAMP WEEK 4: THE SPECTRAL THEOREM
LINEAR ALGEBRA BOOT CAMP WEEK 4: THE SPECTRAL THEOREM Unless otherwise stated, all vector spaces in this worksheet are finite dimensional and the scalar field F is R or C. Definition 1. A linear operator
More informationSection 6.4. The Gram Schmidt Process
Section 6.4 The Gram Schmidt Process Motivation The procedures in 6 start with an orthogonal basis {u, u,..., u m}. Find the Bcoordinates of a vector x using dot products: x = m i= x u i u i u i u i Find
More informationMath 261 Lecture Notes: Sections 6.1, 6.2, 6.3 and 6.4 Orthogonal Sets and Projections
Math 6 Lecture Notes: Sections 6., 6., 6. and 6. Orthogonal Sets and Projections We will not cover general inner product spaces. We will, however, focus on a particular inner product space the inner product
More informationMATH 221, Spring Homework 10 Solutions
MATH 22, Spring 28  Homework Solutions Due Tuesday, May Section 52 Page 279, Problem 2: 4 λ A λi = and the characteristic polynomial is det(a λi) = ( 4 λ)( λ) ( )(6) = λ 6 λ 2 +λ+2 The solutions to the
More informationDot Products. K. Behrend. April 3, Abstract A short review of some basic facts on the dot product. Projections. The spectral theorem.
Dot Products K. Behrend April 3, 008 Abstract A short review of some basic facts on the dot product. Projections. The spectral theorem. Contents The dot product 3. Length of a vector........................
More information235 Final exam review questions
5 Final exam review questions Paul Hacking December 4, 0 () Let A be an n n matrix and T : R n R n, T (x) = Ax the linear transformation with matrix A. What does it mean to say that a vector v R n is an
More informationLinear Algebra Fundamentals
Linear Algebra Fundamentals It can be argued that all of linear algebra can be understood using the four fundamental subspaces associated with a matrix. Because they form the foundation on which we later
More informationMATH 304 Linear Algebra Lecture 23: Diagonalization. Review for Test 2.
MATH 304 Linear Algebra Lecture 23: Diagonalization. Review for Test 2. Diagonalization Let L be a linear operator on a finitedimensional vector space V. Then the following conditions are equivalent:
More informationRecall: Dot product on R 2 : u v = (u 1, u 2 ) (v 1, v 2 ) = u 1 v 1 + u 2 v 2, u u = u u 2 2 = u 2. Geometric Meaning:
Recall: Dot product on R 2 : u v = (u 1, u 2 ) (v 1, v 2 ) = u 1 v 1 + u 2 v 2, u u = u 2 1 + u 2 2 = u 2. Geometric Meaning: u v = u v cos θ. u θ v 1 Reason: The opposite side is given by u v. u v 2 =
More informationMath 350 Fall 2011 Notes about inner product spaces. In this notes we state and prove some important properties of inner product spaces.
Math 350 Fall 2011 Notes about inner product spaces In this notes we state and prove some important properties of inner product spaces. First, recall the dot product on R n : if x, y R n, say x = (x 1,...,
More informationMATH 240 Spring, Chapter 1: Linear Equations and Matrices
MATH 240 Spring, 2006 Chapter Summaries for Kolman / Hill, Elementary Linear Algebra, 8th Ed. Sections 1.1 1.6, 2.1 2.2, 3.2 3.8, 4.3 4.5, 5.1 5.3, 5.5, 6.1 6.5, 7.1 7.2, 7.4 DEFINITIONS Chapter 1: Linear
More informationSolutions to Final Practice Problems Written by Victoria Kala Last updated 12/5/2015
Solutions to Final Practice Problems Written by Victoria Kala vtkala@math.ucsb.edu Last updated /5/05 Answers This page contains answers only. See the following pages for detailed solutions. (. (a x. See
More informationChapter 6. Orthogonality and Least Squares
Chapter 6 Orthogonality and Least Squares Section 6.1 Inner Product, Length, and Orthogonality Orientation Recall: This course is about learning to: Solve the matrix equation Ax = b Solve the matrix equation
More informationLinear Algebra Massoud Malek
CSUEB Linear Algebra Massoud Malek Inner Product and Normed Space In all that follows, the n n identity matrix is denoted by I n, the n n zero matrix by Z n, and the zero vector by θ n An inner product
More information2. Every linear system with the same number of equations as unknowns has a unique solution.
1. For matrices A, B, C, A + B = A + C if and only if A = B. 2. Every linear system with the same number of equations as unknowns has a unique solution. 3. Every linear system with the same number of equations
More informationDIAGONALIZATION. In order to see the implications of this definition, let us consider the following example Example 1. Consider the matrix
DIAGONALIZATION Definition We say that a matrix A of size n n is diagonalizable if there is a basis of R n consisting of eigenvectors of A ie if there are n linearly independent vectors v v n such that
More informationVectors. Vectors and the scalar multiplication and vector addition operations:
Vectors Vectors and the scalar multiplication and vector addition operations: x 1 x 1 y 1 2x 1 + 3y 1 x x n 1 = 2 x R n, 2 2 y + 3 2 2x = 2 + 3y 2............ x n x n y n 2x n + 3y n I ll use the two terms
More informationMAT2342 : Introduction to Applied Linear Algebra Mike Newman, fall Projections. introduction
MAT4 : Introduction to Applied Linear Algebra Mike Newman fall 7 9. Projections introduction One reason to consider projections is to understand approximate solutions to linear systems. A common example
More informationLINEAR ALGEBRA QUESTION BANK
LINEAR ALGEBRA QUESTION BANK () ( points total) Circle True or False: TRUE / FALSE: If A is any n n matrix, and I n is the n n identity matrix, then I n A = AI n = A. TRUE / FALSE: If A, B are n n matrices,
More informationW2 ) = dim(w 1 )+ dim(w 2 ) for any two finite dimensional subspaces W 1, W 2 of V.
MA322 Sathaye Final Preparations Spring 2017 The final MA 322 exams will be given as described in the course web site (following the Registrar s listing. You should check and verify that you do not have
More information1. General Vector Spaces
1.1. Vector space axioms. 1. General Vector Spaces Definition 1.1. Let V be a nonempty set of objects on which the operations of addition and scalar multiplication are defined. By addition we mean a rule
More information(v, w) = arccos( < v, w >
MA322 F all206 Notes on Inner Products Notes on Chapter 6 Inner product. Given a real vector space V, an inner product is defined to be a bilinear map F : V V R such that the following holds: Commutativity:
More informationftuiowamath2550 Assignment NOTRequiredJustHWformatOfQuizReviewForExam3part2 due 12/31/2014 at 07:10pm CST
me me ftuiowamath2550 Assignment NOTRequiredJustHWformatOfQuizReviewForExam3part2 due 12/31/2014 at 07:10pm CST 1. (1 pt) local/library/ui/eigentf.pg A is n n an matrices.. There are an infinite number
More informationDefinitions for Quizzes
Definitions for Quizzes Italicized text (or something close to it) will be given to you. Plain text is (an example of) what you should write as a definition. [Bracketed text will not be given, nor does
More informationInner products. Theorem (basic properties): Given vectors u, v, w in an inner product space V, and a scalar k, the following properties hold:
Inner products Definition: An inner product on a real vector space V is an operation (function) that assigns to each pair of vectors ( u, v) in V a scalar u, v satisfying the following axioms: 1. u, v
More informationSection 6.1. Inner Product, Length, and Orthogonality
Section 6. Inner Product, Length, and Orthogonality Orientation Almost solve the equation Ax = b Problem: In the real world, data is imperfect. x v u But due to measurement error, the measured x is not
More informationThen x 1,..., x n is a basis as desired. Indeed, it suffices to verify that it spans V, since n = dim(v ). We may write any v V as r
Practice final solutions. I did not include definitions which you can find in Axler or in the course notes. These solutions are on the terse side, but would be acceptable in the final. However, if you
More information(v, w) = arccos( < v, w >
MA322 F all203 Notes on Inner Products Notes on Chapter 6 Inner product. Given a real vector space V, an inner product is defined to be a bilinear map F : V V R such that the following holds: For all v,
More informationThe value of a problem is not so much coming up with the answer as in the ideas and attempted ideas it forces on the would be solver I.N.
Math 410 Homework Problems In the following pages you will find all of the homework problems for the semester. Homework should be written out neatly and stapled and turned in at the beginning of class
More informationThe following definition is fundamental.
1. Some Basics from Linear Algebra With these notes, I will try and clarify certain topics that I only quickly mention in class. First and foremost, I will assume that you are familiar with many basic
More informationRecall the convention that, for us, all vectors are column vectors.
Some linear algebra Recall the convention that, for us, all vectors are column vectors. 1. Symmetric matrices Let A be a real matrix. Recall that a complex number λ is an eigenvalue of A if there exists
More informationA Primer in Econometric Theory
A Primer in Econometric Theory Lecture 1: Vector Spaces John Stachurski Lectures by Akshay Shanker May 5, 2017 1/104 Overview Linear algebra is an important foundation for mathematics and, in particular,
More informationOrthogonal Projection and Least Squares Prof. Philip Pennance 1 Version: December 12, 2016
Orthogonal Projection and Least Squares Prof. Philip Pennance 1 Version: December 12, 2016 1. Let V be a vector space. A linear transformation P : V V is called a projection if it is idempotent. That
More informationLinear Models Review
Linear Models Review Vectors in IR n will be written as ordered ntuples which are understood to be column vectors, or n 1 matrices. A vector variable will be indicted with bold face, and the prime sign
More information5.) For each of the given sets of vectors, determine whether or not the set spans R 3. Give reasons for your answers.
Linear Algebra  Test File  Spring Test # For problems  consider the following system of equations. x + y  z = x + y + 4z = x + y + 6z =.) Solve the system without using your calculator..) Find the
More informationLinear Algebra 2 Spectral Notes
Linear Algebra 2 Spectral Notes In what follows, V is an inner product vector space over F, where F = R or C. We will use results seen so far; in particular that every linear operator T L(V ) has a complex
More informationMA 265 FINAL EXAM Fall 2012
MA 265 FINAL EXAM Fall 22 NAME: INSTRUCTOR S NAME:. There are a total of 25 problems. You should show work on the exam sheet, and pencil in the correct answer on the scantron. 2. No books, notes, or calculators
More informationSolving a system by backsubstitution, checking consistency of a system (no rows of the form
MATH 520 LEARNING OBJECTIVES SPRING 2017 BROWN UNIVERSITY SAMUEL S. WATSON Week 1 (23 Jan through 27 Jan) Definition of a system of linear equations, definition of a solution of a linear system, elementary
More informationThe Gram Schmidt Process
u 2 u The Gram Schmidt Process Now we will present a procedure, based on orthogonal projection, that converts any linearly independent set of vectors into an orthogonal set. Let us begin with the simple
More informationThe Gram Schmidt Process
The Gram Schmidt Process Now we will present a procedure, based on orthogonal projection, that converts any linearly independent set of vectors into an orthogonal set. Let us begin with the simple case
More informationMath 18, Linear Algebra, Lecture C00, Spring 2017 Review and Practice Problems for Final Exam
Math 8, Linear Algebra, Lecture C, Spring 7 Review and Practice Problems for Final Exam. The augmentedmatrix of a linear system has been transformed by row operations into 5 4 8. Determine if the system
More informationGlossary of Linear Algebra Terms. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB
Glossary of Linear Algebra Terms Basis (for a subspace) A linearly independent set of vectors that spans the space Basic Variable A variable in a linear system that corresponds to a pivot column in the
More informationReview Notes for Linear Algebra True or False Last Updated: February 22, 2010
Review Notes for Linear Algebra True or False Last Updated: February 22, 2010 Chapter 4 [ Vector Spaces 4.1 If {v 1,v 2,,v n } and {w 1,w 2,,w n } are linearly independent, then {v 1 +w 1,v 2 +w 2,,v n
More informationMATH Linear Algebra
MATH 304  Linear Algebra In the previous note we learned an important algorithm to produce orthogonal sequences of vectors called the GrammSchmidt orthogonalization process. GrammSchmidt orthogonalization
More informationName: Final Exam MATH 3320
Name: Final Exam MATH 3320 Directions: Make sure to show all necessary work to receive full credit. If you need extra space please use the back of the sheet with appropriate labeling. (1) State the following
More information7. Dimension and Structure.
7. Dimension and Structure 7.1. Basis and Dimension Bases for Subspaces Example 2 The standard unit vectors e 1, e 2,, e n are linearly independent, for if we write (2) in component form, then we obtain
More informationorthogonal relations between vectors and subspaces Then we study some applications in vector spaces and linear systems, including Orthonormal Basis,
5 Orthogonality Goals: We use scalar products to find the length of a vector, the angle between 2 vectors, projections, orthogonal relations between vectors and subspaces Then we study some applications
More information