Linear Algebra Lecture Notes

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MATH10212 Linear Algebra Lecture Notes Last change: 02 June 2017 Textbook Students are strongly advised to acquire a copy of the Text- book: D. C. Lay. Linear Algebra and its Applications. Pearson, 2006. ISBN 0-521-28713-4. Other editions can be used as well; the book is easily avail- able on Amazon (this includes some very cheap used copies) and in Blackwell’s bookshop on campus. These lecture notes should be treated only as indica- tion of the content of the course; the textbook contains detailed explanations, worked out examples, etc. About Homework 1 The Undergraduate Student Handbook 2016/17 says: 1 http://www.maths.manchester.ac.uk/study/undergraduate/information-for-current- students/undergraduatestudenthandbook/teachingandlearning/timetabledclasses/2 As a general rule for each hour in class you should spend two hours on independent study. This will include read- ing lecture notes and textbooks, working on exercises and preparing for coursework and examinations. In respect of this course, MATH10212 Linear Algebra B, this means that students are expected to spend 8 (eight) hours a week in private study of Linear Algebra. Normally, homework assignments will consist of some odd numbered exercises from the sections of the Textbook cov- ered in the lectures up to Wednesday in particular week. The Textbook contains answers to most odd numbered ex- ercises (but the numbering of exercises might change from one edition to another). Homework will be given every week on Wednesday and should be returned to Supervision Classes teachers next week, for marking and discussion at supervision classes on Tuesday and Wednesday.3 Communication The Course Webpage is http://www.maths.manchester.ac.uk/ avb/math10212-Linear-Algebra-B. html The Course Webpage page is updated almost daily, some- times several times a day. Refresh it (and files it is linked to) in your browser, otherwise you may miss the changes. Twitter: https://twitter.com/math10212 Email: Feel free to write to me with questions, etc., at the address borovikmanchester.ac.uk but only from your university e-mail account. Emails from Gmail, Hotmail, etc. automatically go to spam. What is Linear Algebra? It is well-known that the total cost of a purchase of amounts g , g , g of some goods at prices p , p , p , respectively, is 1 2 3 1 2 3 an expression 3 X p g + p g + p g = p g . 1 1 2 2 3 3 i i i=1 Expressions of this kind, a x + + a x 1 1 n n are called linear forms in variables x ,::: , x with coeffi- 1 n cients a ,::: , a . 1 n4  Linear Algebra studies the mathematics of linear forms.  Over the course, we shall develop increasingly compact notation for operations of Linear Algebra. In particular, we shall discover that p g + p g + p g 1 1 2 2 3 3 can be very conveniently written as 2 3 g 1   4 5 p p p g 1 2 3 2 g 3 and then abbreviated 2 3 g 1   T 4 5 p p p g = P G, 1 2 3 2 g 3 where 2 3 2 3 p g 1 1 4 5 4 5 P = p and G = g 2 2 p g 3 3  Physicists use even more short notation and, instead of 3 X p g + p g + p g = p g 1 1 2 2 3 3 i i i=1 write 1 2 3 i p g + p g + p g = p g , 1 2 3 i omitting the summation sign entirely. This particular trick was invented by Albert Einstein, of all people. I do not use “physics” tricks in my lectures, but am prepared to give a few additional lectures to physics students.5  Warning: Increasingly compact notation leads to in- creasingly compact and abstract language.  Unlike, say, Calculus, Linear Algebra focuses more on the development of a special mathematics language rather than on procedures.MATH10212  Linear Algebra B  Lecture 1  Linear systems  Last change: 02 June 2017 6 Lecture 1 Systems of linear equations Lay 1.1 A linear equation in the variables x ,::: , x is an equation 1 n that can be written in the form a x + a x + + a x = b 1 1 2 2 n n where b and the coefficients a ,::: , a are real numbers. 1 n The subscript n can be any natural number. A system of simultaneous linear equations is a collec- tion of one or more linear equations involving the same vari- ables, say x ,::: , x . For example, 1 n x + x = 3 1 2 x x = 1 1 2 We shall abbreviate the words “a system of simultaneous linear equations” just to “a linear system”. A solution of the system is a list (s ,::: , s ) of numbers 1 n that makes each equation a true identity when the values s ,::: , s are substituted for x ,::: , x , respectively. For ex- 1 n 1 n ample, in the system above (2, 1) is a solution. The set of all possible solutions is called the solution set of the linear system. Two linear systems are equivalent if the have the same so- lution set. We shall be use the following elementary operations on systems od simultaneous liner equations: Replacement Replace one equation by the sum of itself and a multiple of another equation. Interchange Interchange two equations.MATH10212  Linear Algebra B  Lecture 1  Linear systems  Last change: 02 June 2017 7 Scaling Multiply all terms in a equation by a nonzero con- stant. Note: The elementary operations are reversible. Theorem: Elementary operations preserve equivalence. If a system of simultaneous linear equations is obtained from another system by elementary operations, then the two systems have the same solution set. We shall prove later in the course that a system of linear equations has either  no solution, or  exactly one solution, or  infinitely many solutions, under the assumption that the coefficients and solutions of the systems are real numbers. A system of linear equations is said to be consistent it if has solutions (either one or infinitely many), and a system in inconsistent if it has no solution. Solving a linear system The basic strategy is to replace one system with an equivalent system (that is, with the same solution set) which is easier to solve.MATH10212  Linear Algebra B  Lecture 1  Linear systems  Last change: 02 June 2017 8 Existence and uniqueness questions  Is the system consistent?  If a solution exist, is it unique? Equivalence of linear systems  When are two linear systems equivalent?MATH10212  Linear Algebra B  Lecture 2  Row reduction and echelon forms  Last change: 02 June 2017 9 Lecture 2 Row reduction and echelon forms Lay 1.2 Matrix notation It is convenient to write coefficients of a linear system in the form of a matrix, a rectangular table. For example, the system x 2x + 3x = 1 1 2 3 x + x = 2 1 2 x + x = 3 2 3 has the matrix of coefficients 2 3 1 2 3 4 5 1 1 0 0 1 1 and the augmented matrix 2 3 1 2 3 1 4 5 1 1 0 2 ; 0 1 1 3 notice how the coefficients are aligned in columns, and how missing coefficients are replaced by 0. The augmented matrix in the example above has 3 rows and 4 columns; we say that it is a 34 matrix. Generally, a matrix with m rows and n columns is called an m n matrix. Elementary row operations Replacement Replace one row by the sum of itself and a multiple of another row. Interchange Interchange two rows.MATH10212  Linear Algebra B  Lecture 2  Row reduction and echelon forms  Last change: 02 June 2017 10 Scaling Multiply all entries in a row by a nonzero constant. The two matrices are row equivalent if there is a sequence of elementary row operations that transforms one matrix into the other. Note:  The row operations are reversible.  Row equivalence of matrices is an equivalence relation on the set of matrices. Theorem: Row Equivalence. If the augmented matrices of two linear systems are row equivalent, then the two systems have the same solution set. A nonzero row or column of a matrix is a row or column which contains at least one nonzero entry. We can now formulate a theorem (to be proven later). Theorem: Equivalence of linear systems. Two linear systems are equivalent if and only if the aug- mented matrix of one of them can be obtained from the augmented matrix of another system by frow operations and insertion / deletion of zero rows.MATH10212  Linear Algebra B  Lecture 3  Solution of Linear Systems  Last change: 02 June 2017 11 Lecture 3 Solution of Linear Systems Lay 1.2 A nonzero row or column of a matrix is a row or column which contains at least one nonzero entry. A leading entry of a row is the leftmost nonzero entry (in a non-zero row). Definition. A matrix is in echelon form (or row echelon form) if it has the following three properties: 1. All nonzero rows are above any row of zeroes. 2. Each leading entry of a row is in column to the right of the leading entry of the row above it. 3. All entries in a column below a leading entry are zeroes. If, in addition, the following two conditions are satisfied, 4. All leading entries are equal 1. 5. Each leading 1 is the only non-zero entry in its column then the matrix is in reduced echelon form. An echelon matrix is a matrix in echelon form. Any non-zero matrix can be row reduced (that, transformed by elementary row operations) into a matrix in echelon form (but the same matrix can give rise to different echelon forms). Examples. The following is a schematic presentation of an echelon matrix: 2 3      4 5 0     0 0 0   and this is a reduced echelon matrix:MATH10212  Linear Algebra B  Lecture 3  Solution of Linear Systems  Last change: 02 June 2017 12 2 3 1 0  0  4 5 0 1  0  0 0 0 1  Theorem 1.2.1: Uniqueness of the reduced echelon form. Each matrix is row equivalent to one and only one reduced echelon form. Definition. A pivot position in a matrix A is a location in A that corresponds to a leading 1 in the reduced echelon form of A. A pivot column is a column of A that contains a pivot position. Example for solving in the lecture (The Row Reduction Algorithm): 2 3 0 2 2 2 2 6 7 1 1 1 1 1 6 7 6 7 1 1 1 3 3 4 5 1 1 1 2 2 A pivot is a nonzero number in a pivot position which is used to create zeroes in the column below it. A rule for row reduction: 1. Pick the leftmost non-zero column and in it the topmost nonzero entry; it is a pivot. 2. Using scaling, make the pivot equal 1. 3. Using replacement row operations, kill all non-zero en- tries in the column below the pivot. 4. Mark the row and column containing the pivot as piv- oted.MATH10212  Linear Algebra B  Lecture 3  Solution of Linear Systems  Last change: 02 June 2017 13 5. Repeat the same with the matrix made of not pivoted yet rows and columns. 6. When this is over, interchange the rows making sure that the resulting matrix is in echelon form. 7. Using replacement row operations, kill all non-zero en- tries in the column above the pivot entries. Solution of Linear Systems When we converted the augmented matrix of a linear sys- tem into its reduced row echelon form, we can write out the entire solution set of the system. Example. Let 2 3 1 0 5 1 4 5 0 1 1 4 0 0 0 0 be the augmented matrix of a a linear system; then the sys- tem is equivalent to x 5x = 1 1 3 x + x = 4 2 3 0 = 0 The variables x and x correspond to pivot columns in the 1 2 matrix and a re called basic variables (also leading or pivot variables). The other variable, x is a free variable. 3 Free variables can be assigned arbitrary values and then leading variables expressed in terms of free variables: x = 1 + 5x 1 3 x = 4 x 2 3 x is free 3MATH10212  Linear Algebra B  Lecture 3  Solution of Linear Systems  Last change: 02 June 2017 14 Theorem 1.2.2: Existence and Uniqueness A linear system is consistent if and only if the rightmost col- umn of the augmented matrix is not a pivot column—that is, if and only if an echelon form of the augmented matrix has no row of the form   with b nonzero 0  0 b If a linear system is consistent, then the solution set con- tains either (i) a unique solution, when there are no free variables, or (ii) infinitely many solutions, when there is at least one free variable. Using row reduction to solve a linear system 1. Write the augmented matrix of the system. 2. Use the row reduction algorithm to obtain an equivalent augmented matrix in echelon form. Decide whether the system is consistent. 3. if the system is consistent, get the reduced echelon form. 4. Write the system of equations corresponding to the ma- trix obtained in Step 3. 5. Express each basic variable in terms of any free vari- ables appearing in the equation.MATH10212  Linear Algebra B  Lecture 4  Vector equations  Last change: 02 June 2017 15 Lecture 4 Vector equations Lay 1.3 A matrix with only one column is called a column vector, or simply a vector. n R is the set of all column vectors with n entries. A row vector: a matric with one row. Two vectors are equal if and only if they have  the same shape,  the same number of rows,  and their corresponding entries are equal. n The set of al vectors with n entries is denotedR . n The sum u + v of two vectors u and v inR is obtained by 2 adding corresponding entries in u and v. For example inR       1 1 0 + = . 2 1 1 The scalar multiple cv of a vector v and a real number (“scalar”) c is the vector obtained by multiplying each entry 3 in v by c. For example inR , 2 3 2 3 1 1.5 4 5 4 5 1.5 0 = 0 . 3 2 The vector whose entries are all zeroes is called the zero vector and denoted 0: 2 3 0 6 7 0 6 7 0 =6 7 . . . 4 .5 0MATH10212  Linear Algebra B  Lecture 4  Vector equations  Last change: 02 June 2017 16 Operations with row vectors are defined in a similar way. n Algebraic properties ofR n For all u, v, w2R and all scalars c and d: 1. u + v = v + u 2. (u + v) + w = u + (v + w) 3. u + 0 = 0 + u = u 4. u + (u) =u + u = 0 5. c(u + v) = cu + cv 6. (c + d)u = cu + du 7. c(du) = (cd)u 8. 1u = u (Hereu denotes (1)u.) Linear combinations n Given vectors v , v ,::: , v inR and scalars c , c ,::: , c , 1 2 p 1 2 p the vector y = c v + c v 1 1 p p is called a linear combination of v , v ,::: , v with weights 1 2 p c , c ,::: , c . 1 2 pMATH10212  Linear Algebra B  Lecture 4  Vector equations  Last change: 02 June 2017 17 Rewriting a linear system as a vector equation Consider an example: the linear system x + x = 2 2 3 x + x + x = 3 1 2 3 x + x x = 2 1 2 3 can be written as equality of two vectors: 2 3 2 3 x + x 2 2 3 4 5 4 5 x + x + x = 3 1 2 3 x + x x 2 1 2 3 which is the same as 2 3 2 3 2 3 2 3 0 1 1 2 4 5 4 5 4 5 4 5 x + x + x = 1 1 1 3 1 2 3 1 1 1 2 Let us write the matrix 2 3 0 1 1 2 4 5 1 1 1 3 1 1 1 2 in a way that calls attention to its columns:   a a a b 1 2 3 Denote 2 3 2 3 2 3 0 1 1 4 5 4 5 4 5 a = 1 , a = 1 , a = 1 1 2 3 1 1 1 and 2 3 2 4 5 b = 3 , 2MATH10212  Linear Algebra B  Lecture 4  Vector equations  Last change: 02 June 2017 18 then the vector equation can be written as x a + x a + x a = b. 1 1 2 2 3 3 Notice that to solve this equation is the same as express b as a linear combination of a , a , a , and 1 2 3 find all such expressions. Therefore solving a linear system is the same as finding an expres- sion of the vector of the right part of the system as a linear combination of columns in its matrix of coefficients. A vector equation x a + x a + + x a = b. 1 1 2 2 n n has the same solution set as the linear system whose aug- mented matrix is   a a  a b 1 2 n In particular b can be generated by a linear combination of a , a ,::: , a if and only if there is a solution of the corre- 1 2 n sponding linear system. n Definition. If v ,::: , v are inR , then the set of all linear 1 p combination of v ,::: , v is denoted by 1 p Spanfv ,::: , vg 1 p n and is called the subset ofR spanned (or generated) by v ,::: , v ; or the span of vectors v ,::: , v . 1 p 1 p That is, Spanfv ,::: , vg is the collection of all vectors which 1 p can be written in the form c v + c v + + c v 1 1 2 2 p pMATH10212  Linear Algebra B  Lecture 4  Vector equations  Last change: 02 June 2017 19 with c ,::: , c scalars. 1 p n We say that vectors v ,::: , v spanR if 1 p n Spanfv ,::: , vg =R 1 p We can reformulate the definition of span as follows: iv n a ,::: , a 2R , then 1 p n Spanfa ,::: , ag = fb2R such that 1 p a  ajb 1 p is the augmented matrix of a consistent system of linear equationsg, or, in simpler words, n Spanfa ,::: , ag = fb2R such that 1 p the system of equations x a + + x a = b 1 1 p p has a solutiong.MATH10212  Linear Algebra B  Lecture 5  The matrix equation Ax = b  Last change: 02 June 2017 20 Lecture 5: The matrix equation Ax = b Lay 1.4 Definition. If A is an m n matrix, with columns a ,::: , a , 1 n n and if x is in R , then the product of A and x, denoted Ax, is the linear combination of the columns of A using the corresponding entries in x as weights: 2 3 x 1   . 4 .5 Ax = a a  a . 1 2 n x n = x a + x a + + x a 1 1 2 2 n n Example. The system x + x = 2 2 3 x + x + x = 3 1 2 3 x + x x = 2 1 2 3 was written as x a + x a + x a = b. 1 1 2 2 3 3 where 2 3 2 3 2 3 0 1 1 4 5 4 5 4 5 a = 1 , a = 1 , a = 1 1 2 3 1 1 1 and 2 3 2 4 5 b = 3 . 2 In the matrix product notation it becomes 2 32 3 2 3 0 1 1 x 2 1 4 54 5 4 5 1 1 1 x = 3 2 1 1 1 x 2 3

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