1 MAC 2103 Module 10 lnner Product Spaces I

2 Rev.F09 Learning Objectives Upon completing this module, you should be able to: 1. Define and find the inner product, norm and distance in a given inner product space. 2. Find the cosine of the angle between two vectors and determine if the vectors are orthogonal in an inner product space. 3. Find the orthogonal complement of a subspace of an inner product space. 4. Find a basis for the orthogonal complement of a subspace of ℜ ⁿ spanned by a set of row vectors. 5. Identify the four fundamental matrix spaces of an m x n matrix A, and know the column rank of A, the row rank of A, and the rank of A. 6. Know the equivalent statements of an n x n matrix A. Click link to download other modules.

3 Rev.09 General Vector Spaces II Click link to download other modules. Inner Product Spaces, Inner Products, Norm, Distance, Fundamental Matrix Spaces, Rank, and Orthogonality The major topics in this module:

4 Rev.F09 Definition of an Inner Product and Orthogonal Vectors Click link to download other modules. A. Inner Product: Let be any function from into that satisfies the following conditions for any u, v, z in V: and iff Then defines an inner product for all u, v in V. u and v in V are Orthogonal Vectors iff

5 Rev.F09 Definition of a Norm of a Vector Click link to download other modules. B. Norm: Let be any function from into that satisfies the following conditions for any u, v in V: Then defines a norm for all u in V. The special norm that is induced by an inner product is

6 Rev.F09 Definition of the Distance Between Two Vectors Click link to download other modules. C. Distance: Let be any function from into that satisfies the following conditions for any u, v, w in V: Then defines the distance for all u, v in V. The special distance that is induced by a norm is

7 Rev.F09 How to Find an Inner Product, Norm, and Distance for Vectors in ℜ ⁿ? Click link to download other modules. For u and v in ℜ ⁿ, we define and Note: These are our usual norm and distance in ℜ ⁿ. ℜ ⁿ is an inner product space with the dot product as its inner product.

8 Rev.F09 How to Find an Inner Product, Norm, and Distance for Vectors in ℜ ⁿ? (Cont.) Click link to download other modules. Example 1: Let u = (1,2,-1) and v = (2,0,1) in ℜ ³. Find the inner product, norms, and distance.

9 Rev.F09 How to Find an Inner Product, Norm, and Distance for Vectors (Matrices) in M 22 ? Click link to download other modules. In M 22, the vectors are 2 x 2 matrices. Thus, given two matrices U and V in M 22, we define Recall: tr(A) = sum of the entries on the main diagonal of A.

10 Rev.F09 How to Find an Inner Product, Norm, and Distance for Vectors (Matrices) in M 22 ? (Cont.) Click link to download other modules. Example 2: Let be matrices in the vector space M 22. Find the inner product, norms and distance for U and V in M 22. Recall: tr(A) = sum of the entries on the main diagonal of A.

11 Rev.F09 How to Find an Inner Product, Norm, and Distance for Vectors (Matrices) in M 22 ? (Cont.) Click link to download other modules.

12 Rev.F09 How to Find an Inner Product, Norm, and Distance for Vectors (Polynomials) in P 2 ? Click link to download other modules. Let P 2 be the set of all polynomials of degree less than or equal to two. For two polynomials we define

13 Rev.F09 How to Find an Inner Product, Norm, and Distance for Vectors (Polynomials) in P 2 ? (Cont.) Click link to download other modules. Example 3: Find the inner product, norms, and distance for p and q in P 2, where

14 Rev.F09 How to Find an Inner Product, Norm, and Distance for Vectors (Functions) in C[a,b]? Click link to download other modules. Let C[a,b] be the set of all continuous functions on [a,b]. For all f and g in C[a,b], we define

15 Rev.F09 How to Find an Inner Product, Norm, and Distance for Vectors (Functions) in C[a,b]? (Cont.) Click link to download other modules. Example 4: Let f = f(x) = 1 and g = g(x) = -x. Find the inner product, norms, and distance on C[2,4].

16 Rev.F09 The Cauchy-Schwarz Inequality Click link to download other modules. Cauchy-Schwarz Inequality: for all f, g in an inner product space V, with the norm induced by the inner product. We evaluate the Cauchy-Schwarz inequality for the previous examples.

17 Rev.F09 Orthogonality for Vectors (Functions) in C[a,b] Click link to download other modules. Example 5: Let f = f(x) = 1 and g = g(x) = e x. Find the inner product on C[0,2]. Since the inner product is not zero, f and g are not orthogonal functions in C[0,2]. Example 6: Let f = f(x) = 5 and g = g(x) = cos(x). Find the inner product on C[0,π]. Since the inner product is zero, f and g are orthogonal functions in C[0,π].

18 Rev.F09 Orthogonality for Vectors (Functions) in C[a,b] (Cont.) Click link to download other modules. Example 7: Let f = f(x) = 4x and g = g(x) = x 2. Find the inner product on C[-1,1]. Since the inner product is zero, f and g are orthogonal functions in C[-1,1].

19 Rev.F09 Inner Product and Orthogonality for Vectors (Functions) in C[a,b] (Cont.) Click link to download other modules. Example 8: Let f = f(x) = 1 and g = g(x) = sin(2x). Find the inner product on C[-π,π]. Since the inner product is zero, f and g are orthogonal functions in C[-π,π].

20 Rev.F09 The Generalized Theorem of Pythogoras Click link to download other modules. Generalized Theorem of Pythogoras: If f and g are orthogonal vectors in an inner product space V, then Proof: Let f and g be orthogonal in V, then

21 Rev.F09 The Generalized Theorem of Pythogoras (Cont.) Click link to download other modules. The Generalized Theorem of Pythogoras We evaluate the theorem for the previous examples. From example 6, we have

22 Rev.F09 The Generalized Theorem of Pythogoras (Cont.) Click link to download other modules. From Example 7, we have Notice that we have used orthogonality to obtain our results without directly computing the integrals. Next, we will compute the integrals directly to obtain the result.

23 Rev.F09 The Generalized Theorem of Pythogoras (Cont.) Click link to download other modules. Thus,

24 Rev.F09 How to Find the Cosine of the Angle Between Two Vectors in an Inner Product Space? Click link to download other modules. Example 9: Let ℜ⁵ have the Euclidean inner product. Find the cosine of the angle between u = (2,1,0,-1,5) and v = (1,-1,-2,3,1).

25 Rev.F09 How to Find the Cosine of the Angle Between Two Vectors in an Inner Product Space? (Cont.) Click link to download other modules. Example 10: Let P 2 have the previous inner product. Find the cosine of the angle between p and q.

26 Rev.F09 Quick Review on a Basis for a Vector Space and the Dimension of a Vector Space Click link to download other modules. Let be nonzero vectors in V. Then for, we have that W = span(S) = {all linear combinations of } is a subspace of V. If S contains some linearly dependent vectors, then the dim(W) < n. If S is a linearly independent set of vectors, then S is a basis for the vector space W. W is a proper subspace of V, if dim(W) < dim(V). If dim(W) = dim(V) = n, then S is a basis for V. A non-trivial vector space V always have two subspaces: the trivial vector space, {0} and V.

27 Rev.F09 Quick Review on a Basis for a Vector Space and the Dimension of a Vector Space Click link to download other modules. So, we need two conditions for the set S to be a basis of a vector space, it must be a linearly independent set and it must span the vector space. The number of basis vectors in S is the dimension of the vector space. For example: dim( ℜ ²)=2, dim( ℜ ³)=3, dim( ℜ⁵ )=5, dim (P 2 )=3, dim(M 22 )=4, and dim(C[a,b]) is infinite.

28 Rev.F09 Rank, Row Rank, and Column Rank of a Matrix Click link to download other modules. The dimensions of the row space of A, column space of A, and nullspace of A are also called the row rank of A, column rank of A, and nullity of A, respectively. The rank of A = the row rank of A = the column rank of A, and is the number of linearly independent columns in the matrix A. The nullity of A = dim(null(A)) and is the number of free variables in the solution space.

29 Rev.F09 The Orthogonal Complement of a Subspace W Click link to download other modules. The orthogonal complement of W, is the set of all vectors in a finite dimensional inner product space V that are orthogonal to every vector in W. Both W and are subspaces of V where dim(W) + dim( ) = dim(V). The only vector common to W and is 0, and

30 Rev.F09 The Four Fundamental Matrix Spaces Click link to download other modules. Let A be an m x n matrix, then: 1.The null(A) and the row(A) are orthogonal complements in ℜ ⁿ with respect to the Euclidean inner product. 2.The null(A T ) and the col(A) are orthogonal complements in ℜ with respect to the Euclidean inner product. Note that row(A) = col(A T ) and col(A) = row(A T ). The four fundamental matrix spaces are: 1. row(A) = col(A T ), 2. col(A) = row(A T ), 3. null(A), and 4. null(A T ).

31 Rev.F09 How to Find a Basis for the Orthogonal Complement of the Subspace of ℜ ⁿ Spanned by the Vectors? Click link to download other modules. Example 11: Find a basis for the orthogonal complement of the subspace of ℜ ⁿ spanned by the vectors v 1, v 2, and v 3. Let V = span({v 1, v 2, v 3 }). Then, V is a subspace of of ℜ ⁿ if v 1, v 2, v 3 ∈ ℜ ⁿ. Let From the last module, we know that the null(A) is the orthogonal complement to the row(A) = col(A T ), and that the null(A) is the solution space of the homogeneous system, Ax = 0. Let v 1, v 2, v 3 be the row vectors of the matrix A. Then, row(A) is the orthogonal complement to null(A) which is the solution space to Ax = 0. We shall use Gauss Elimination to reduce A to a row echelon form to find the basis for null(A).

32 Rev.F09 How to Find a Basis for the Orthogonal Complement of the Subspace of ℜ ⁿ Spanned by the Vectors? (Cont.) Click link to download other modules. Since the null(A) is the solution space of the homogeneous system, Ax = 0, the general solution of the system is: Note that column 1 and column 2, with the red leading 1’s, are linearly independent. The corresponding columns of A form a basis for the col(A). Then, dim(col(A)) = 2.

33 Rev.F09 How to Find a Basis for the Orthogonal Complement of the Subspace of ℜ ⁿ Spanned by the Vectors? (Cont.) Click link to download other modules. The nonzero rows with the leading 1’s are linearly independent. These rows form a basis for row(A). Then, dim(row(A)) = 2. Note that the row space and column space are both two dimensional, so rank(A) = 2. The rank(A) is the common dimension of the row space of A and the column space of A. So, rank(A) = rowrank(A) = colrank(A) = dim(col(A)) =dim(row(A)) is always true for any m x n matrix A.

34 Rev.F09 How to Find a Basis for the Orthogonal Complement of the Subspace of ℜ ⁿ Spanned by the Vectors? (Cont.) Click link to download other modules. Thus, the solution vector is: and the vector {w 1 } forms a basis for the null(A). In other words, {w 1 } = {(16,19,1)} forms a basis for the orthogonal complement of row(A).

35 Rev.F09 How to Find a Basis for the Orthogonal Complement of the Subspace of ℜ ⁿ Spanned by the Vectors? (Cont.) Click link to download other modules. Since B = {w 1 } is a basis for null(A), span(B) = null(A), and dim(null(A)) = 1 is the nullity of A. In this example, rank(A) is 2 and the nullity of A is 1, and the number of columns of A is 3. For any m x n matrix A, rank(A) + nullity(A) is n, which is the number of columns of A.

36 Rev.F09 Let’s Review Some Equivalent Statements If A is an n x n matrix, and if T A : ℜ ⁿ→ ℜ ⁿ is multiplication by A, than the following are equivalent. (See Theorem 6.2.7) A is invertible. Ax = 0 has only the trivial solution The reduced row-echelon form of A is I n. A is expressible as a product of elementary matrices. Ax = b is consistent for every n x 1 matrix b. Ax = b has exactly one solution for every n x 1 matrix b. det(A) ≠ 0. The range of T A is ℜ ⁿ. T A is one-to-one. Click link to download other modules.

37 Rev.F09 Let’s Review Some Equivalent Statements (Cont.) The column vectors of A are linearly independent. The row vectors of A are linearly independent. The column vectors of A span ℜ ⁿ. The row vectors of A span ℜ ⁿ. The column vectors of A form a basis for ℜ ⁿ. The row vectors of A form a basis for ℜ ⁿ. A has rank n = row rank n = column rank n = dim(col(A)). A has nullity 0 = dim(null(A)). The orthogonal complement of the nullspace of A, null(A), is ℜ ⁿ. The orthogonal complement of the row space of A, row(A), is {0}. Click link to download other modules.

38 Rev.F09 What have we learned? We have learned to: 1. Define and find the inner product, norm and distance in a given inner product space. 2. Find the cosine of the angle between two vectors and determine if the vectors are orthogonal in an inner product space. 3. Find the orthogonal complement of a subspace of an inner product space. 4. Find a basis for the orthogonal complement of a subspace of ℜ ⁿ spanned by a set of row vectors. 5. Identify the four fundamental matrix spaces of an m x n matrix A, and know the column rank of A, the row rank of A, and the rank of A. 6. Know the equivalent statements of an n x n matrix A. Click link to download other modules.

39 Rev.F09 Credit Some of these slides have been adapted/modified in part/whole from the following textbook: Anton, Howard: Elementary Linear Algebra with Applications, 9th Edition Click link to download other modules.