INNER PRODUCTS. 1. Real Inner Products Definition 1. An inner product on a real vector space V is a function (u, v) u, v from V V to R satisfying


 Collin Dalton
 1 years ago
 Views:
Transcription
1 INNER PRODUCTS 1. Real Inner Products Definition 1. An inner product on a real vector space V is a function (u, v) u, v from V V to R satisfying (1) αu + βv, w = α u, w + β v, s for all u, v, w V and all α, β R; (2) u, v = v, u for all u, v V ; (3) v, v 0 for all v V, with equality if and only if v = 0. A real vector space together with an inner product is called a real inner product space. Item 1 asserts that the map (u, v) u, v is linear in the first variable. In view of 2, it is also linear in the second variable. Example 2. The usual dot product defines an inner product on R n. u, v = u v = u i v i Example 3. The set C([a, b]) of all continuous real valued functions on the interval [0, 1] forms a real vector space under pointwise addition and scaling. We can define an inner product on C([a, b]) by f, g = b a f(x)g(x) dx. Definition 4. Let V be a real vector space with inner product,. For any v V we define the norm of v by v = v, v 1/2. Note that the square root on the right is defined as a nonnegative real number by 3. Lemma 5. Let V be a real vector space with inner product,. (1) (Positivity) v 0, with equality if and only if v = 0. (2) (Homogeneity) αv = α v for every α R and every v V. (3) (CauchySchwarz Inequality) u, v u v, with equality if and only if u and v are parallel. (4) (Triangle Inequality) u + v u + v. Proof. Positivity is immediate from item 3 in the definition of inner product. For homogeneity, we have αv 2 = αv, αv = α 2 v, v = α 2 v 2. The result follows by taking square roots. 1
2 INNER PRODUCTS 2 We now turn to the CauchySchwarz Inequality. If u and v are parallel, then one is a scalar multiple of the other, say v = αu for some α R. Therefore u, v = u, αu = α u, u = α u 2 = u αu = u v. On the other hand, if u and v are not parallel, then u + αv 0 for every α R, so, by positivity, 0 < u + αv 2 = u + αv, u + αv = u u, v α + v 2 α 2 In particular, taking α = u,v v 2 gives 0 < u 2 u, v 2 v 2, from which the CauchySchwarz Inequality follows. Finally, we turn to the proof of the Triangle Inequality. We have u + v 2 = u + v, u + v = u u, v + v 2 u u, v + v 2 u 2 + u v + v 2 = ( u + v ) 2. Taking square roots gives the desired result. Definition 6. Two vectors u and v in a real inner product space are said to be orthogonal if u, v = 0. Theorem 7 (Pythagorean Theorem). If u and v are orthogonal vectors in a real inner product space, then Proof. u + v 2 = u 2 + v 2. u + v 2 = u + v, u + v = u u, v + v 2 = u 2 + v Exercises. (1) When does equality hold in the Triangle Inequality? (2) For any nonnegative integer n, define c n, s n C([0, 2π]) by c n (x) = cos nx, s n (x) = sin nx. Using the inner product of Example 3: (a) Calculate c n and s n. (b) Show that c n and s m are orthogonal for every m, n Z +. (c) Show that c n is orthogonal to c m and s n is orthogonal to x m whenever n m. (3) Prove the converse of the Pythagorean Theorem: In a real inner product space, if u + v 2 = u 2 + v 2, then u and v are orthogonal.
3 INNER PRODUCTS 3 2. The Complex Case It is tempting to define an inner product on a complex vector space by simply transcribing the definition of a real inner product, but allowing the scalars to be complex. Unfortunately, this does not lead to anything useful: There would be no inner products on any complex vector space containing a nonzero vector (see Exercise 1). A slight refinement of the second item in the definition is needed. Definition 8. An inner product on a complex vector space V is a function (u, v) u, v from V V to C satisfying (1) αu + βv, w = α u, w + β v, s for all u, v, w V and all α, β C; (2) u, v = v, u for all u, v V ; (3) v, v 0 for all v V, with equality if and only if v = 0. A complex vector space together with an inner product is called a complex inner product space. We will use to phrase inner product space to refer to either a real or complex inner product space. Note that a complex inner product is linear in the first variable, but not in the second. In fact, we have u, αv + βw = αv + βw, u = α v, u + β w, u = α v, u + β w, u = α u, v + β u, w. We say that, is conjugate linear in the second variable. Example 9. The Hermitian inner product on C n is defined by z, w = z i w i. As in the real case, we define v = v, v 1/2, and we say u and v are orthogonal if u, v = 0. Complex variants of Lemma 5 and Theorem 7 can be established as in the real case. We state them here for completeness. The proofs are left to the exercises. Lemma 10. Let V be a complex vector space with inner product,. (1) (Positivity) v 0, with equality if and only if v = 0. (2) (Homogeneity) αv = α v for every α C and every v V. (3) (CauchySchwarz Inequality) u, v u v, with equality if and only if u and v are parallel. (4) (Triangle Inequality) u + v u + v. Theorem 11 (Pythagorean Theorem). If u and v are orthogonal vectors in a complex inner product space, then u + v 2 = u 2 + v 2.
4 INNER PRODUCTS Exercises. (1) Show that if the conjugation on the right side of 2 were omitted no inner products would exist on any complex vector space containing a nonzero vector. (2) Let V be a complex vector space. We may view V as a real vector space by simply ignoring nonreal scalars. Now suppose that, is a complex inner product on V, and define u, v = Re u, v. (a) Show that this defines a real inner product. (b) Show that the real inner product, and the complex inner product, define the same norm. (c) Show that orthogonality with respect to the complex inner product implies orthogonality with respect to the real inner product, but not conversely. It follows that the real Pythagorean Theorem is stronger than the complex version. (3) Let, be an inner product on a complex vector space V. (a) Show that for every u, v V and every α C we have u + αv = u Re (α u, v ) + α 2 v 2 (b) Prove the CauchySchwarz Inequality for complex inner products. Hint: Take α = u,v v Orthonormal Sets Definition 12. A set of vectors {e 1,..., e n } in an inner product space is said to be orthogonal if e i, e j = 0 whenever i j. If, in addition, e i = 1 for each i, we call the set orthonormal. An orthonormal set which is also a basis for V is an orthonormal basis. Theorem 13. If {e 1,..., e n } is an orthonormal basis for V, then for every v V we have v = v, e i e i. Corollary 14. If {e 1,..., e n } is an orthonormal basis for V, then for any u, v V we have u, v = u, e i v, e i. Note that the corollary asserts that the inner product of two vectors in a real or complex finite dimensional inner product space is the Euclidean or, respectively, Hermitian inner product of their coordinate vectors in R n or C n. Corollary 15. Let T be a linear transformation from a finite dimensional inner product space V to a finite dimensional inner product space W. If A = [a ij ] is the matrix of T with respect to ordered orthonormal bases (e 1,..., e n ) and (f 1,..., f m ), respectively, then a ij = T e j, e i. Proof. The jth column of is the coordinate vector of T e j with respect to the basis (f 1,..., f m ). The result follows from Corollary 13.
5 INNER PRODUCTS 5 Theorem 16 (Bessel s Inequality). If {e 1,..., e n } is orthonormal in V, then for every v V we have v, e i 2 v 2. Moreover, we have equality for every v V if and only if the set is an orthonormal basis. Theorem 17. Let {v 1,..., v n } be linearly independent. There is an orthonormal set {e 1,..., e n } such that for 1 k n we have span{e 1,..., e k } = span{v 1,..., v k }. Proof (GramSchmidt Process): Let V k = span{v 1,..., v k }. We first construct, by induction on k, an orthogonal set {w 1,..., w n } such that span{w 1,..., w k } = V k. The required orthonormal set is then obtained by setting e k = w k w k. We begin by setting w 1 = v 1. For 1 < k n, suppose by induction that we have an orthogonal set {w 1,..., w k 1 } which spans V k 1. Since dim V k 1 = k 1, it follows that {w 1,..., w k 1 } is a basis for V k 1, and in particular, w i 0 for 1 i k 1. Let k 1 w k = v k v k, w i w i 2 w i. It follows from the orthogonality of {w 1,..., w k 1 } that w k, w i = 0 for 1 i k 1, and further that w k 0, since v k V k 1. Moreover, w k V k, so {w 1,..., w k } is a linearly independent set in V k. Since dim V k = k, it must be a basis for V k. This completes the proof. Corollary 18. Let V be an inner product space of finite dimension n, and let {e 1,..., e k } be orthonormal in V. There are e k+1,..., e n V such that {e 1,..., e n } is an orthonormal basis. In particular, every finite dimensional inner product space has an orthonormal basis. 4. Orthogonal Projections Definition 19. Let V be an inner product space and let E V. The orthogonal complement of E is Lemma 20. E is a subspace of V. E = {v V : v, w = 0 for every w E}. Theorem 21 (Projection Theorem). Let V be an inner product space and let W be a finite dimensional subspace. For every v V there are unique v 1, v 2 V such that (1) v 1 W and v 2 W ; (2) v = v 1 + v 2. Proof. Let {e 1,..., e k } be an orthonormal basis for W. If v 1 and v 2 are as advertised, then v, e i = v 1 + v 2, e i = v 1, e i + v 2, e i = v 1, e i so k k v, e i e i = v 1, e i e i = v 1
6 INNER PRODUCTS 6 so v 1, and hence also v 2, is uniquely determined. To complete the proof, let v V, and define k (1) v 1 = v, e i e i and v 2 = v v 1. Clearly v 1 W and v = v 1 + v 2. It only remains to check that v 2 W. It is straightforward to check that v 2 is orthogonal to each e i, and therefore to any linear combination of {e 1,..., e k }. Since {e 1,..., e k } is a basis for W, it follows that v 2 is orthogonal to each member of W, which completes the proof. Remark 22. The vector v 1 in the Projection Theorem is the orthogonal projection of v on the subspace W. The proof shows that v 1 is given by (1), where {e 1,..., e k } can be any orthonormal basis for the subspace W. The linear operator defined by k (2) P v = v, e i e i is the orthogonal projector of V onto W. The uniqueness assertion in the Projection Theorem assures us that the operator P defined by (2) does not depend on the choice of an orthonormal basis for W. Corollary 23. If W is a subspace of a finite dimensional inner product space V, then dim W + dim W = dim V. Proof. If {e 1,..., e m } is an orthonormal basis for W and {f 1,..., f n } is an orthonormal basis for W, then {e 1,..., e m, f 1,..., f n } is an orthonormal set which spans V, and is therefore an orthonormal basis for V. Definition 24. A linear functional on a real (or complex) vector space V is a linear transformation from V into R (or C). Theorem 25 (Representation Theorem for Linear Functionals). Let λ be a linear functional on a finite dimensional inner product space V. There is a unique w V such that λ(v) = v, w for every v V. Proof. We first establish uniqueness. Suppose that v, w 1 = v, w 2 for every v V. Then v, w 1 w 2 = 0 for every v V. In particular, taking v = w 1 w 2 we obtain w 1 w 2 2 = 0, and hence w 1 = w 2, establishing uniqueness. For existence, since V is finite dimensional, it has an orthonormal basis {e 1,..., e n }. For every v V we have v = v, e i e i so Λv = v, e i Λe i = v, (Λe i )e i = v, w
7 INNER PRODUCTS 7 with w = (Λe i )e i. 5. Adjoints Let V and W be finite dimensional inner product spaces over the same scalar field (R or C), and let T : V W be linear. Our immediate goal is to define a transposed transformation T going in the reverse direction, from W to V. Lemma 26. Let V, W, and T be as above. For each w W there is a unique w V such that for every v V we have T v, w = v, w. Proof. Apply the Representation Theorem to the linear functional λ(v) = T v, w. Definition 27. With V, W, and T as above, we define the adjoint mapping T : W V by defining T w to be the unique w V such that T v, w = v, w for every v V. Thus the map T is characterized by the identity for all v V and w W. Lemma 28. T is linear. T v, w = v, T w Proof. Let w 1, w 2 W and let α 1, α 2 be scalars. Then for every v V we have v, T (α 1 w 1 + α 2 w 2 ) = T v, α 1 w 1 + α 2 w 2 Since this holds for every v V, it follows that Theorem 29. (1) (T ) = T (2) (S + T ) = S + T (3) (αt ) = αt (4) (ST ) = T S = α 1 T v, w 1 + α 2 T v, w 2 = α 1 v, T w 1 + α 2 v, T w 2 = v, α 1 T w 1 + α 2 T w 2 T (α 1 w 1 + α 2 w 2 ) = α 1 T w 1 + α 2 T w 2. Theorem 30. Let V and W be finite dimensional inner product spaces and let T be a linear transformation from V to W. Let A be the matrix of T with respect to ordered orthonormal bases. Then the matrix of T is A T. This is an immediate consequence of Corollary 15 and the definition of the adjoint transformation. Of course, in the real case, the bar over A can be omitted.
8 INNER PRODUCTS 8 Definition 31. A linear operator T on a finite dimensional inner product space is selfadjoint if T = T. A real n n matrix that defines a self adjoint operator on R n is called real symmetric. A complex n n matrix that defines a selfadjoint operator on C n is Hermitian. Note that a real n n matrix is real symmetric if and only if A T = A. A complex n n matrix is Hermitian if and only if A T = A. Lemma 32. All eigenvalues of a selfadjoint operator are real. Proof. Let α be an eigenvalue of a selfadjoint operator with eigenvector v. Then so α = α. α v 2 = α v, v = αv, v = T v, v = v, T v = v, αv = α v 2 Note that we do not (yet) assert that eigenvalues exist. Corollary 33. Every Hermitian matrix has a real eigenvalue. Proof. An n n Hermitian matrix defines a self adjoint operator on C n, which has a complex eigenvalue by the Fundamental Theorem of Algebra. By the previous lemma, this complex eigenvalue must be real. Corollary 34. Every selfadjoint operator has a real eigenvalue. Proof. In the complex case, this is immediate from Lemma 32 and the Fundamental Theorem of Algebra. For the real case, it suffices to show that the matrix A of the transformation with respect to some orthonormal basis has a real eigenvalue. But the matrix is real symmetric, and hence also Hermitian, so, by the previous corollary, it has a real eigenvalue. 6. Orthogonal and Unitary Operators Theorem 35 (Polarization Identities). (1) If V is a real inner product space, then for every u, v V we have u, v = 1 ( u + v 2 u 2 v 2). 2 (2) If V is a complex inner product space, then for every u, v V we have u, v = 1 ( u + v 2 + i u + iv 2 u v 2 i u iv 2). 4 Proof. For any scalar α we have (3) u + αv 2 = u + αv, u + αv = u 2 + α u, v + α v, u + α 2 v 2 In the real case, we have v, u = u, v, and the result follows by taking α = 1. For the complex case, note that when α = 1, (3) becomes, upon multiplication by α α u + αv 2 = α u 2 + u, v + α 2 u, v + α v 2. In particular, taking α = 1, 1, i, and i gives u + v 2 = u 2 + u, v + v, u + v 2 u + v 2 = u 2 u, v + v, u v 2 i u + iv 2 = i u 2 + u, v v, u + i v 2 i u iv 2 = i u 2 + u, v v, u i v 2
9 INNER PRODUCTS 9 Adding the four identities above gives the desired result. Theorem 36. Let T be a linear operator on a finite dimensional vector space V. The following are equivalent. (1) For every u, v V we have T u, T v = u, v. (2) For every v V we have T v = v. (3) For every orthonormal basis {e 1,..., e n } for V, the set {T e 1,..., T e n } is also an orthonormal basis for V. (4) For some orthonormal basis {e 1,..., e n } for V, the set {T e 1,..., T e n } is also an orthonormal basis for V. (5) T T is the identity operator on V. (6) T T is the identity operator on V. Proof. The implication 1 = 2 is trivial, and the reverse implication is immediate from the Polarization Identity. The implications 1 = 3 = 4 are also trivial. 4 = 1: Let {e 1,..., e n } be an orthonormal basis. For u, v V we can write so u = α 1 e α n e n v = β 1 e β n e n T u = α 1 T e α n T e n T v = β 1 T e β n T e n. Since {T e 1,..., T e n } is an orthonormal basis, we have T u, T v = α i β i = u, v. We have now established the equivalence of 1 2. The equivalence of 5 and 6 is a consequence of the general fact that one sided inverses for linear operators on finite dimensional vector spaces are two sided inverses. We complete the proof by establishing equivalence of 5 and 1. 1 = 5: Let u V. Then for any v V we have T T u, v = T u, T v = u, v Since this is true for every v V, it follows that T T u = u. Since u V is arbitrary, it follows that T T is the identity operator on V. 5 = 1: If T T is the identity operator, then for any u, v V we have T u, T v = T T u, v = u, v. A linear operator on a real inner product space satisfying one, and hence all, of the conditions of Theorem 36 is called orthogonal. A linear operator on a complex inner product space satisfying one, and hence all, of the conditions of Theorem 36 is called unitary.
10 INNER PRODUCTS Spectral Theorem In this section, we identify the linear operators on a finite dimensional inner product space V which can be diagonalized by an orthogonal (in the real case) or unitary (in the complex case) transformation. Equivalently, we want to identify the linear the linear operators T on V such that V has an orthonormal basis consisting of eigenvectors for T. Suppose that T is a linear transformation on an inner product space V, and that {e 1,..., e n } is an orthonormal basis for V consisting of eigenvectors of T, with corresponding eigenvalues λ 1,..., λ n. Then T e j, e i = e j, T e i = e j, λ i e i = λ i δ ij. It follows that T e j = λ j e j. Therefore, T T e j = T T e j = λ j 2 e j. Since the operators T T and T T agree on a basis, we have T T = T T. Definition 37. A linear operator T on a finite dimensional inner product space is normal if T T = T T. For example, orthogonal, unitary, symmetric, and Hermitian operators are all normal. We have established the following. Lemma 38. If T is a linear operator on an inner product space V which admits an orthonormal basis consisting of eigenvectors for T, then T is normal. The Spectral Theorem asserts that the converse holds for linear operators on complex inner product spaces. Theorem 39 (Spectral Theorem). Let T be a normal operator on a finite dimensional complex inner product space V. Then V has an orthonormal basis consisting of eigenvectors for T. The proof of the Spectral Theorem requires some preliminary lemmas. Lemma 40. If T is a normal operator on V, then T v = T v for every v V. Proof. T v 2 = T v, T v = T T v, v = T T v, v = T v, T v = T v 2. Corollary 41. If T is normal then T and T have the same null space. Corollary 42. Let T be a normal operator with eigenvalue λ, and corresponding eigenvector v. Then λ is an eigenvalue for T with the same eigenvector v. Proof. Since T is normal, so is T λi, so T λi and (T λi) = T λi have the same null space. Lemma 43. Let T be a normal operator on V with eigenvalue λ. Let W be the corresponding eigenspace. Then W is invariant under T. Proof. Let v W. Then for any w W we have so T v W. T v, w = v, T w = v, λw = λ v, w = 0
11 INNER PRODUCTS 11 Proof of the Spectral Theorem. We use induction on the dimension n of V. If n = 1, then any unit vector in V forms an orthonormal basis. For n > 1, suppose by induction that the conclusion holds for every complex inner product space of dimension less than n. By the Fundamental Theorem of Algebra, the characteristic polynomial of T has a complex root λ, which must be an eigenvalue of T. Let W be the corresponding eigenspace, and let {e 1,..., e k } be an orthonormal basis for W. By Lemma 43, V 0 = W is invariant under T, so the restriction T 0 of T to V 0 is a linear operator on V 0. Further, the restriction, 0 of the inner product to V 0 V 0 is an inner product on V 0. One easily verifies that T0 is the restriction of T to V 0, and so T 0 is normal. Since the dimension of V 0 is less than n, it follows from our induction hypothesis that V 0 has an orthonormal basis {f 1,..., f l } consisting of eigenvalues for T 0. It follows that {e 1,..., e k, f 1,..., f l } is an orthonormal basis for V The real case. The conclusion of the Spectral Theorem may fail for normal operators on real inner product spaces for the simple reason that such operators need not have eigenvalues. For example, a rotation of the plane through an angle which is not a multiple of pi is normal but has no eigenvalues. Lemma 44. Let T be a linear operator on a finite dimensional real inner product space V. If V has an orthonormal basis consisting of eigenvectors of T, then T is selfadjoint. Proof. Let {e 1,..., e n } be an orthonormal basis of V consisting of eigenvectors for T, with corresponding eigenvalues λ 1,..., λ n. Arguing as in the proof of Lemma 38, we obtain T e j = λ n e j = T e j, so T and T agree on a bases, and are therefore equal. The real version of the Spectral Theorem asserts the converse. Theorem 45 (Real Spectral Theorem). If T is a self adjoint operator on a finite dimensional real inner product space V, then V has an orthonormal basis consisting of eigenvectors for T. The proof of the real version of the Spectral Theorem is essentially the same as the proof of the complex version, appealing to Corollary 34 instead of the Fundamental Theorem of Algebra for the existence of eigenvalues.
Inner Product Spaces and Orthogonality
Inner Product Spaces and Orthogonality week 34 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 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 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 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 SelfAdjoint and Normal Operators
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 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 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 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 informationSolutions to Linear Algebra Practice Problems
Solutions to Linear Algebra Practice Problems. Find all solutions to the following systems of linear equations. (a) x x + x 5 x x x + x + x 5 (b) x + x + x x + x + x x + x + 8x Answer: (a) We create 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 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 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 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 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 informationNotes on Jordan Canonical Form
Notes on Jordan Canonical Form Eric Klavins University of Washington 8 Jordan blocks and Jordan form A Jordan Block of size m and value λ is a matrix J m (λ) having the value λ repeated along the main
More information1 Eigenvalues and Eigenvectors
Math 20 Chapter 5 Eigenvalues and Eigenvectors Eigenvalues and Eigenvectors. Definition: A scalar λ is called an eigenvalue of the n n matrix A is there is a nontrivial solution x of Ax = λx. Such an x
More information(January 14, 2009) End k (V ) End k (V/W )
(January 14, 29) [16.1] Let p be the smallest prime dividing the order of a finite group G. Show that a subgroup H of G of index p is necessarily normal. Let G act on cosets gh of H by left multiplication.
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 yaxis We observe that
More informationInner products on R n, and more
Inner products on R n, and more Peyam Ryan Tabrizian Friday, April 12th, 2013 1 Introduction You might be wondering: Are there inner products on R n that are not the usual dot product x y = x 1 y 1 + +
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 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 understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied,
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 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 Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 19967 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More informationChapter 17. Orthogonal Matrices and Symmetries of Space
Chapter 17. Orthogonal Matrices and Symmetries of Space Take a random matrix, say 1 3 A = 4 5 6, 7 8 9 and compare the lengths of e 1 and Ae 1. The vector e 1 has length 1, while Ae 1 = (1, 4, 7) has length
More informationNOTES on LINEAR ALGEBRA 1
School of Economics, Management and Statistics University of Bologna Academic Year 205/6 NOTES on LINEAR ALGEBRA for the students of Stats and Maths This is a modified version of the notes by Prof Laura
More informationLinear Algebra Notes for Marsden and Tromba Vector Calculus
Linear Algebra Notes for Marsden and Tromba Vector Calculus ndimensional Euclidean Space and Matrices Definition of n space As was learned in Math b, a point in Euclidean three space can be thought of
More informationInner Product Spaces. 7.1 Inner Products
7 Inner Product Spaces 71 Inner Products Recall that if z is a complex number, then z denotes the conjugate of z, Re(z) denotes the real part of z, and Im(z) denotes the imaginary part of z By definition,
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 informationFinite dimensional C algebras
Finite dimensional C algebras S. Sundar September 14, 2012 Throughout H, K stand for finite dimensional Hilbert spaces. 1 Spectral theorem for selfadjoint opertors Let A B(H) and let {ξ 1, ξ 2,, ξ n
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 informationSec 4.1 Vector Spaces and Subspaces
Sec 4. Vector Spaces and Subspaces Motivation Let S be the set of all solutions to the differential equation y + y =. Let T be the set of all 2 3 matrices with real entries. These two sets share many common
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 ndimensional 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 informationTHE DIMENSION OF A VECTOR SPACE
THE DIMENSION OF A VECTOR SPACE KEITH CONRAD This handout is a supplementary discussion leading up to the definition of dimension and some of its basic properties. Let V be a vector space over a field
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 informationWHICH LINEARFRACTIONAL TRANSFORMATIONS INDUCE ROTATIONS OF THE SPHERE?
WHICH LINEARFRACTIONAL TRANSFORMATIONS INDUCE ROTATIONS OF THE SPHERE? JOEL H. SHAPIRO Abstract. These notes supplement the discussion of linear fractional mappings presented in a beginning graduate course
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 information1. True/False: Circle the correct answer. No justifications are needed in this exercise. (1 point each)
Math 33 AH : Solution to the Final Exam Honors Linear Algebra and Applications 1. True/False: Circle the correct answer. No justifications are needed in this exercise. (1 point each) (1) If A is an invertible
More informationVector Spaces II: Finite Dimensional Linear Algebra 1
John Nachbar September 2, 2014 Vector Spaces II: Finite Dimensional Linear Algebra 1 1 Definitions and Basic Theorems. For basic properties and notation for R N, see the notes Vector Spaces I. Definition
More information1. Let P be the space of all polynomials (of one real variable and with real coefficients) with the norm
Uppsala Universitet Matematiska Institutionen Andreas Strömbergsson Prov i matematik Funktionalanalys Kurs: F3B, F4Sy, NVP 0050615 Skrivtid: 9 14 Tillåtna hjälpmedel: Manuella skrivdon, Kreyszigs bok
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 realvalued matrix A is said to be an orthogonal
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 informationNumerical Analysis Lecture Notes
Numerical Analysis Lecture Notes Peter J. Olver 5. Inner Products and Norms The norm of a vector is a measure of its size. Besides the familiar Euclidean norm based on the dot product, there are a number
More informationBANACH AND HILBERT SPACE REVIEW
BANACH AND HILBET SPACE EVIEW CHISTOPHE HEIL These notes will briefly review some basic concepts related to the theory of Banach and Hilbert spaces. We are not trying to give a complete development, but
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 informationLinear Algebra. A vector space (over R) is an ordered quadruple. such that V is a set; 0 V ; and the following eight axioms hold:
Linear Algebra A vector space (over R) is an ordered quadruple (V, 0, α, µ) such that V is a set; 0 V ; and the following eight axioms hold: α : V V V and µ : R V V ; (i) α(α(u, v), w) = α(u, α(v, w)),
More informationChapter 20. Vector Spaces and Bases
Chapter 20. Vector Spaces and Bases In this course, we have proceeded stepbystep through lowdimensional Linear Algebra. We have looked at lines, planes, hyperplanes, and have seen that there is no limit
More informationLecture 1: Schur s Unitary Triangularization Theorem
Lecture 1: Schur s Unitary Triangularization Theorem This lecture introduces the notion of unitary equivalence and presents Schur s theorem and some of its consequences It roughly corresponds to Sections
More informationWe call this set an ndimensional parallelogram (with one vertex 0). We also refer to the vectors x 1,..., x n as the edges of P.
Volumes of parallelograms 1 Chapter 8 Volumes of parallelograms In the present short chapter we are going to discuss the elementary geometrical objects which we call parallelograms. These are going to
More informationMATH 304 Linear Algebra Lecture 24: Scalar product.
MATH 304 Linear Algebra Lecture 24: Scalar product. Vectors: geometric approach B A B A A vector is represented by a directed segment. Directed segment is drawn as an arrow. Different arrows represent
More information1 Sets and Set Notation.
LINEAR ALGEBRA MATH 27.6 SPRING 23 (COHEN) LECTURE NOTES Sets and Set Notation. Definition (Naive Definition of a Set). A set is any collection of objects, called the elements of that set. We will most
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 informationLEARNING OBJECTIVES FOR THIS CHAPTER
CHAPTER 2 American mathematician Paul Halmos (1916 2006), who in 1942 published the first modern linear algebra book. The title of Halmos s book was the same as the title of this chapter. FiniteDimensional
More informationLinear Algebra Problems
Math 504 505 Linear Algebra Problems Jerry L. Kazdan Note: New problems are often added to this collection so the problem numbers change. If you want to refer others to these problems by number, it is
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 information4. MATRICES Matrices
4. MATRICES 170 4. Matrices 4.1. Definitions. Definition 4.1.1. A matrix is a rectangular array of numbers. A matrix with m rows and n columns is said to have dimension m n and may be represented as follows:
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 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 information4: EIGENVALUES, EIGENVECTORS, DIAGONALIZATION
4: EIGENVALUES, EIGENVECTORS, DIAGONALIZATION STEVEN HEILMAN Contents 1. Review 1 2. Diagonal Matrices 1 3. Eigenvectors and Eigenvalues 2 4. Characteristic Polynomial 4 5. Diagonalizability 6 6. Appendix:
More informationSection 1.1. Introduction to R n
The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to
More informationPresentation 3: Eigenvalues and Eigenvectors of a Matrix
Colleen Kirksey, Beth Van Schoyck, Dennis Bowers MATH 280: Problem Solving November 18, 2011 Presentation 3: Eigenvalues and Eigenvectors of a Matrix Order of Presentation: 1. Definitions of Eigenvalues
More informationC 1 x(t) = e ta C = e C n. 2! A2 + t3
Matrix Exponential Fundamental Matrix Solution Objective: Solve dt A x with an n n constant coefficient matrix A x (t) Here the unknown is the vector function x(t) x n (t) General Solution Formula in Matrix
More informationState of Stress at Point
State of Stress at Point Einstein Notation The basic idea of Einstein notation is that a covector and a vector can form a scalar: This is typically written as an explicit sum: According to this convention,
More informationLecture 18  Clifford Algebras and Spin groups
Lecture 18  Clifford Algebras and Spin groups April 5, 2013 Reference: Lawson and Michelsohn, Spin Geometry. 1 Universal Property If V is a vector space over R or C, let q be any quadratic form, meaning
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 informationLEARNING OBJECTIVES FOR THIS CHAPTER
CHAPTER 6 Woman teaching geometry, from a fourteenthcentury edition of Euclid s geometry book. Inner Product Spaces In making the definition of a vector space, we generalized the linear structure (addition
More informationLecture 2: Essential quantum mechanics
Department of Physical Sciences, University of Helsinki http://theory.physics.helsinki.fi/ kvanttilaskenta/ p. 1/46 Quantum information and computing Lecture 2: Essential quantum mechanics JaniPetri Martikainen
More informationFinite Dimensional Hilbert Spaces and Linear Inverse Problems
Finite Dimensional Hilbert Spaces and Linear Inverse Problems ECE 174 Lecture Supplement Spring 2009 Ken KreutzDelgado Electrical and Computer Engineering Jacobs School of Engineering University of California,
More information13 MATH FACTS 101. 2 a = 1. 7. The elements of a vector have a graphical interpretation, which is particularly easy to see in two or three dimensions.
3 MATH FACTS 0 3 MATH FACTS 3. Vectors 3.. Definition We use the overhead arrow to denote a column vector, i.e., a linear segment with a direction. For example, in threespace, we write a vector in terms
More information3 Orthogonal Vectors and Matrices
3 Orthogonal Vectors and Matrices The linear algebra portion of this course focuses on three matrix factorizations: QR factorization, singular valued decomposition (SVD), and LU factorization The first
More informationOctober 3rd, 2012. Linear Algebra & Properties of the Covariance Matrix
Linear Algebra & Properties of the Covariance Matrix October 3rd, 2012 Estimation of r and C Let rn 1, rn, t..., rn T be the historical return rates on the n th asset. rn 1 rṇ 2 r n =. r T n n = 1, 2,...,
More informationDiagonalisation. Chapter 3. Introduction. Eigenvalues and eigenvectors. Reading. Definitions
Chapter 3 Diagonalisation Eigenvalues and eigenvectors, diagonalisation of a matrix, orthogonal diagonalisation fo symmetric matrices Reading As in the previous chapter, there is no specific essential
More informationx1 x 2 x 3 y 1 y 2 y 3 x 1 y 2 x 2 y 1 0.
Cross product 1 Chapter 7 Cross product We are getting ready to study integration in several variables. Until now we have been doing only differential calculus. One outcome of this study will be our ability
More informationMATH 110 Spring 2015 Homework 6 Solutions
MATH 110 Spring 2015 Homework 6 Solutions Section 2.6 2.6.4 Let α denote the standard basis for V = R 3. Let α = {e 1, e 2, e 3 } denote the dual basis of α for V. We would first like to show that β =
More informationThe determinant of a skewsymmetric matrix is a square. This can be seen in small cases by direct calculation: 0 a. 12 a. a 13 a 24 a 14 a 23 a 14
4 Symplectic groups In this and the next two sections, we begin the study of the groups preserving reflexive sesquilinear forms or quadratic forms. We begin with the symplectic groups, associated with
More informationA Modern Course on Curves and Surfaces. Richard S. Palais
A Modern Course on Curves and Surfaces Richard S. Palais Contents Lecture 1. Introduction 1 Lecture 2. What is Geometry 4 Lecture 3. Geometry of InnerProduct Spaces 7 Lecture 4. Linear Maps and the Euclidean
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 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 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 ndimensional column
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 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 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 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 information4.5 Linear Dependence and Linear Independence
4.5 Linear Dependence and Linear Independence 267 32. {v 1, v 2 }, where v 1, v 2 are collinear vectors in R 3. 33. Prove that if S and S are subsets of a vector space V such that S is a subset of S, then
More informationLinear Algebra I. Ronald van Luijk, 2012
Linear Algebra I Ronald van Luijk, 2012 With many parts from Linear Algebra I by Michael Stoll, 2007 Contents 1. Vector spaces 3 1.1. Examples 3 1.2. Fields 4 1.3. The field of complex numbers. 6 1.4.
More informationBindel, Spring 2012 Intro to Scientific Computing (CS 3220) Week 3: Wednesday, Feb 8
Spaces and bases Week 3: Wednesday, Feb 8 I have two favorite vector spaces 1 : R n and the space P d of polynomials of degree at most d. For R n, we have a canonical basis: R n = span{e 1, e 2,..., e
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. September 23, 2010
LINEAR ALGEBRA September 3, 00 Contents 0. LUdecomposition.................................... 0. Inverses and Transposes................................. 0.3 Column Spaces and NullSpaces.............................
More informationINTRODUCTORY LINEAR ALGEBRA WITH APPLICATIONS B. KOLMAN, D. R. HILL
SOLUTIONS OF THEORETICAL EXERCISES selected from INTRODUCTORY LINEAR ALGEBRA WITH APPLICATIONS B. KOLMAN, D. R. HILL Eighth Edition, Prentice Hall, 2005. Dr. Grigore CĂLUGĂREANU Department of Mathematics
More informationDeterminants, Areas and Volumes
Determinants, Areas and Volumes Theodore Voronov Part 2 Areas and Volumes The area of a twodimensional object such as a region of the plane and the volume of a threedimensional object such as a solid
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 informationTv = Tu 1 + + Tu N. (u 1,...,u M, v 1,...,v N ) T(Sv) = S(Tv) = Sλv = λ(sv).
54 CHAPTER 5. EIGENVALUES AND EIGENVECTORS 5.6 Solutions 5.1 Suppose T L(V). Prove that if U 1,...,U M are subspaces of V invariant under T, then U 1 + + U M is invariant under T. Suppose v = u 1 + + u
More informationNumerical Analysis Lecture Notes
Numerical Analysis Lecture Notes Peter J. Olver 6. Eigenvalues and Singular Values In this section, we collect together the basic facts about eigenvalues and eigenvectors. From a geometrical viewpoint,
More informationMatrix Norms. Tom Lyche. September 28, Centre of Mathematics for Applications, Department of Informatics, University of Oslo
Matrix Norms Tom Lyche Centre of Mathematics for Applications, Department of Informatics, University of Oslo September 28, 2009 Matrix Norms We consider matrix norms on (C m,n, C). All results holds for
More informationSolutions to Review Problems
Chapter 1 Solutions to Review Problems Chapter 1 Exercise 42 Which of the following equations are not linear and why: (a x 2 1 + 3x 2 2x 3 = 5. (b x 1 + x 1 x 2 + 2x 3 = 1. (c x 1 + 2 x 2 + x 3 = 5. (a
More informationLinear Codes. In the V[n,q] setting, the terms word and vector are interchangeable.
Linear Codes Linear Codes In the V[n,q] setting, an important class of codes are the linear codes, these codes are the ones whose code words form a subvector space of V[n,q]. If the subspace of V[n,q]
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 51 Orthonormal
More information2. Introduction to quantum mechanics
2. Introduction to quantum mechanics 2.1 Linear algebra Dirac notation Complex conjugate Vector/ket Dual vector/bra Inner product/bracket Tensor product Complex conj. matrix Transpose of matrix Hermitian
More informationMetric Spaces. Chapter 7. 7.1. Metrics
Chapter 7 Metric Spaces A metric space is a set X that has a notion of the distance d(x, y) between every pair of points x, y X. The purpose of this chapter is to introduce metric spaces and give some
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 information