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 with an m n matrix A, its column space, null space, and row space. March 26, 2018 1 / 52
Theorem If two matrices A and B are row equivalent, then their row spaces are the same. In particular, if B is an echelon form of the matrix A, then the nonzero rows of B form a basis for Row B and also for Row A since these are the same space. March 26, 2018 2 / 52
Example A matrix A along with its rref is shown. 2 5 8 0 17 A = 1 3 5 1 5 3 11 19 7 1 1 7 13 5 3 1 0 1 0 1 0 1 2 0 3 0 0 0 1 5 0 0 0 0 0 (a) Find a basis for Row A and state the dimension dim Row A. March 26, 2018 3 / 52
Example continued... (b) Find a basis for Col A and state its dimension. March 26, 2018 4 / 52
Example continued... (c) Find a basis for Nul A and state its dimension. March 26, 2018 5 / 52
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Remarks We can naturally associate three vector spaces with an m n matrix A. Row A and Nul A are subspaces of R n and Col A is a subspace of R m. Careful! The rows of the rref do span Row A. But we go back to the columns in the original matrix to get vectors that span Col A. (Get a basis for Col A from A itself!) Careful Again! Just because the first three rows of the rref span Row A does not mean the first three rows of A span Row A. (Get a basis for Row A from the rref!) March 26, 2018 7 / 52
Remarks Row operations preserve row space, but change linear dependence relations of rows. Row operations change column space, but preserve linear dependence relations of columns. Another way to obtain a basis for Row A is to take the transpose A T and do row operations. We have the following relationships: Col A = Row A T and Row A = Col A T. The dimension of the null space is called the nullity. March 26, 2018 8 / 52
Rank Definition: The rank of a matrix A (denoted rank A) is the dimension of the column space of A. Theorem: For m n matrix A, dim Col A = dim Row A = rank A. Moreover rank A + dim Nul A = n. Note: This theorem states the rather obvious fact that { } { } { number of number of total number + = pivot columns non-pivot columns of columns }. March 26, 2018 9 / 52
Examples (1) A is a 5 4 matrix with rank A = 4. What is dim Nul A? (2) If A is 7 5 and dim Col A = 2. Determine the nullity 1 of A and rank A T. March 26, 2018 10 / 52
Addendum to Invertible Matrix Theorem Let A be an n n matrix. The following are equivalent to the statement that A is invertible. (m) The columns of A form a basis for R n (n) Col A = R n (o) dim Col A = n (p) rank A = n (q) Nul A = {0} (r) dim Nul A = 0 March 26, 2018 11 / 52
Section 6.1: Inner Product, Length, and Orthogonality Recall: A vector u in R n can be considered an n 1 matrix. It follows that u T is a 1 n matrix u T = [u 1 u 2 u n ]. Definition: For vectors u and v in R n we define the inner product of u and v (also called the dot product) by the matrix product v 1 u T v 2 v = [u 1 u 2 u n ]. = u 1v 1 + u 2 v 2 + + u n v n. v n Note that this product produces a scalar. It is sometimes called a scalar product. March 26, 2018 12 / 52
Theorem (Properties of the Inner Product) We ll use the notation u v = u T v. Theorem: For u, v and w in R n and real scalar c (a) u v = v u (b) (u + v) w = u w + v w (c) c(u v) = (cu) v = u (cv) (d) u u 0, with u u = 0 if and only if u = 0. March 26, 2018 13 / 52
The Norm The property u u 0 means that u u always exists as a real number. Definition: The norm of the vector v in R n is the nonnegative number denoted and defined by v = v v = v1 2 + v 2 2 + + v n 2 where v 1, v 2,..., v n are the components of v. As a directed line segment, the norm is the same as the length. March 26, 2018 14 / 52
Norm and Length Figure: In R 2 or R 3, the norm corresponds to the classic geometric property of length. March 26, 2018 15 / 52
Unit Vectors and Normalizing Theorem: For vector v in R n and scalar c cv = c v. Definition: A vector u in R n for which u = 1 is called a unit vector. Remark: Given any nonzero vector v in R n, we can obtain a unit vector u in the same direction as v u = v v. This process, of dividing out the norm, is called normalizing the vector v. March 26, 2018 16 / 52
Example Show that v/ v is a unit vector. March 26, 2018 17 / 52
Example Find a unit vector in the direction of v = (1, 3, 2). March 26, 2018 18 / 52
Distance in R n Definition: For vectors u and v in R n, the distance between u and v is denoted and defined by dist(u, v) = u v. Example: Find the distance between u = (4, 0, 1, 1) and v = (0, 0, 2, 7). March 26, 2018 19 / 52
Orthogonality Definition: Two vectors are u and v orthogonal if u v = 0. Figure: Note that two vectors are perpendicular if u v = u + v March 26, 2018 20 / 52
Orthogonal and Perpendicular Show that u v = u + v if and only if u v = 0. March 26, 2018 21 / 52
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The Pythagorean Theorem Theorem: Two vectors u and v are orthogonal if and only if u + v 2 = u 2 + v 2. March 26, 2018 24 / 52
Orthogonal Complement Definition: Let W be a subspace of R n. A vector z in R n is said to be orthogonal to W if z is orthogonal to every vector in W. Definition: Given a subspace W of R n, the set of all vectors orthogonal to W is called the orthogonal complement of W and is denoted by W. March 26, 2018 25 / 52
Theorem: W is a subspace of R n. March 26, 2018 26 / 52
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Example Let W =Span 1 0 0, 0 0 1. Show that W =Span 0 1 0. Give a geometric interpretation of W and W as subspaces of R 3. March 26, 2018 28 / 52
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Example [ 1 3 2 Let A = 2 0 4 [Row(A)]. ]. Show that if x is in Nul(A), then x is in March 26, 2018 31 / 52
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Theorem Theorem: Let A be an m n matrix. The orthogonal complement of the row space of A is the null space of A. That is [Row(A)] = Nul(A). The orthongal complement of the column space of A is the null space of A T i.e. [Col(A)] = Nul(A T ). March 26, 2018 34 / 52
Example: Find the orthogonal complement of Col(A) A = 5 2 1 3 3 0 2 4 1 2 2 9 0 1 1 March 26, 2018 35 / 52
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Section 6.2: Orthogonal Sets Remark: We know that if B = {b 1,..., b p } is a basis for a subspace W of R n, then each vector x in W can be realized (uniquely) as a sum x = c 1 b 2 + + c p b p. If n is very large, the computations needed to determine the coefficients c 1,..., c p may require a lot of time (and machine memory). Question: Can we seek a basis whose nature simplifies this task? And what properties should such a basis possess? March 26, 2018 37 / 52
Orthogonal Sets Definition: An indexed set {u 1,..., u p } in R n is said to be an orthogonal set provided each pair of distinct vectors in the set is orthogonal. That is, provided u i u j = 0 whenever i j. Example: Show that the set orthogonal set. 3 1 1, 1 2 1, 1 4 7 is an March 26, 2018 38 / 52
3 1 1, 1 2 1, 1 4 7 March 26, 2018 39 / 52
Orthongal Basis Definition: An orthogonal basis for a subspace W of R n is a basis that is also an orthogonal set. Theorem: Let {u 1,..., u p } be an orthogonal basis for a subspace W of R n. Then each vector y in W can be written as the linear combination y = c 1 u 1 + c 2 u 2 + + c p u p, where the weights c j = y u j u j u j. March 26, 2018 40 / 52
Example 3 1 1, 1 2 1 the vector y =, 2 3 0 1 4 7 is an orthogonal basis of R3. Express as a linear combination of the basis vectors. March 26, 2018 41 / 52
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Projection Given a nonzero vector u, suppose we wish to decompose another nonzero vector y into a sum of the form y = ŷ + z in such a way that ŷ is parallel to u and z is perpendicular to u. March 26, 2018 43 / 52
Projection Since ŷ is parallel to u, there is a scalar α such that ŷ = αu. March 26, 2018 44 / 52
Projection onto the subspace L =Span{u} Notation: ŷ = proj L = [ ] [ 7 4 Example: Let y = and u = 6 2 Span{u} and z is orthogonal to u. ( y u ) u u u ]. Write y = ŷ + z where ŷ is in March 26, 2018 45 / 52
Example Continued... Determine the distance between the point (7, 6) and the line Span{u}. March 26, 2018 46 / 52
Orthonormal Sets Definition: A set {u 1,..., u p } is called an orthonormal set if it is an orthogonal set of unit vectors. Definition: An orthonormal basis of a subspace W of R n is a basis that is also an orthonormal set. Example: Show that R 2. 3 5 4 5, 4 5 3 5 is an orthonormal basis for March 26, 2018 47 / 52
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Orthogonal Matrix [ 3 Consider the matrix U = 5 4 ] 5 whose columns are the vectors in the last example. Compute the product 4 5 3 5 U T U What does this say about U 1? March 26, 2018 49 / 52
Orthogonal Matrix Definition: A square matrix U is called an orthogonal matrix if U T = U 1. Theorem: An n n matrix U is orthogonal if and only if it s columns form an orthonormal basis of R n. The linear transformation associated to an orthogonal matrix preserves lenghts and angles in the following sense: March 26, 2018 50 / 52
Theorem: Orthogonal Matrices Let U be an n n orthogonal matrix and x and y vectors in R n. Then (a) Ux = x (b) (Ux) (Uy) = x y, in particular (c) (Ux) (Uy) = 0 if and only if x y = 0. Proof (of (a)): March 26, 2018 51 / 52
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