1300 Linear Algebra and Vector Geometry

Transcription

1 1300 Linear Algebra and Vector Geometry R. Craigen Office: MH May-June 2017

2 Introduction: linear equations Read 1.1 (in the text that is!) Go to course, class webpages. Familiarize yourself their contents complete syllabus information (to appear) and link to Math 1300 Wiki Do all suggested homework problems from 1.1 (see course web page!) What is a linear equation? An equation (in 2 variables) whose graph is a line. (Get it? Linear!) ax + by = c Not all linear equations have this form (EG y = mx + b) but every line is expressed by some equation of this form a and b are... constants, called the coefficients (of variables x, y respectively)

3 Linear equations in many variables A linear equation in three variables is one of the form (or can be put in this form ) ax + by + cz = d Or... (since we re soon going to run out of names for variables and coefficients) a 1 x 1 + a 2 x 2 + a 3 x 3 = b By extension a linear equation in n variables (or unknowns) x 1,... x n with coefficients a 1,..., a n, b (b is called the constant coefficient) is one of the form a 1 x 1 + a 2 x a n x n = b

4 Linear systems: systems of linear equations A system of m linear equations in variables/unknowns x 1,..., x n is a collection of equations of the form a 11 x 1 + a 12 x a 1n x n = b 1 a 21 x 1 + a 22 x a 2n x n = b 2.. a m1 x 1 + a m2 x a mn x n = b m Observe: The double index for a ij tells you which equation it is in (i) and which variable it is a coefficient for (j) Notice we also index the constant coefficients (to indicate which equation) EG: x + y z = 3 x + 2y + 3z = 5 is a system of how many equations? 2 in how many variables? 3 In our notation, what is a 22? 2 a 13? 1

5 Solving equations An equation like 2x y + z = 5 is a statement about the variables x, y, z. It can be true (EG if (x, y, z) = (1, 1, 4)) or false (EG if (x, y, z) = (0, 0, 0)). A solution of an equation is an assignment of values making the equation true. The solution of an equation is the set of all solutions. To solve an equation means to show its solution. (i.e., all solutions). A parameter is a symbol representing a quantity that is an unrestricted choice of any (real) number. We use parameters to display infinitely many solutions. To solve this equation set x = s, y = t (s, t are parameters). Then z = 5 2x + y = 5 2s + t And our solution is (x, y, z) = (s, t, 5 2s + t).

6 Solving systems x + 5y 3z = 2 A system of equations, like y + 2z = 3 is a collection of one or more statements (i.e., equations). A solution to a system is any assignment of values making all of its equations true. The solution is the set of all solutions. EG: (x, y, z) = (0, 1, 1) is a solution to this system. But not (x, y, z) = (3, 3, 0) though it makes one equation true! To solve the above system subtract 5 (Eq. 2) from (Eq. 1): x + 5y 3z = 2 y + 2z = 3 = x 13z = 13 y + 2z = 3 Observe x, y are easily solved in terms of z. Think of z as free we can set it equal to a parameter: z = s. Our solution: (x, y, z) = (13s 13, 3 2s, s)

7 How many solutions can a system have? A system of linear equations can have: No solutions (We say such a system is inconsistent) EG: x + 2y + 3z = 1 2x + 4y + 6z = 0 Else it is consistent in one of two ways: exactly one solution EG: x + 2y = 1 x + y = 3 = y = 2 x + y = 3 = (x, y) = (5, 2) or infinitely many solutions EG: x + 2y = 1 x + 2y = 1 = = (x, y) = (1 2s, s) 2x + 4y = 2 0 = 0 No other possibilities!

8 Geometrical meaning of the three types Points common to all graphs of equations in the system. In the plane (x, y): intersection of lines In (3D) space (x, y, z): intersection of planes

9 Operations for manipulating systems of equations Operations which do not change the set of solutions Basic trick: perform operations that change the system but do not change the set of solutions 1. Multiply (both sides of) an equation by a nonzero constant EG: x + 2y = 1 x + 2y = 1 x 2y = 1 2x + 6y = 4 x + 3y = 2 x + 3y = 2 2. Interchange two equations x + y + z = 1 x + 2y + 3z = 2 x + z = 0 x + z = 0 x + 2y + 3z = 2 x + y + z = 1 3. Add a constant multiple of one equation to another x + y + z = 1 2x + 3y + 4z = 5 x + y + z = 1 Add ( 2) y + 2z = 3 first Eq. to second Eq.

10 The augmented matrix of a system Doing those operations on a system is TEDIOUS! Too many symbols to move around. (Unnecessarily). The augmented matrix of system a 11 x 1 + a 12 x a 1n x n = b a m1 x 1 + a m2 x a mn x n = b m Is the rectangular array of numbers a 11 a 12 a 1n b a m1 a m2 a mn b m All necessary information for a system is in its augmented matrix Variables are represented by columns We can reconstruct any system from its augmented matrix.

11 Elementary Row Operations (EROs) Operations on systems EROs Our operations on systems act only on the coefficients the data in its augmented matrix. They correspond to: 1. Multiply a row of the matrix by any nonzero constant ( 1 2 ) 1 ( 1 2 ) 1 ( R R 2 R 1 R 1 ) 2. Interchange two rows R 1 R Add any constant multiple of one row to another row ( ) 1 ( ) R 2 R 2 2R 1

12 Row Echelon form (REF) 1.2 We have EROs to manipulate systems of equations......but how to use them to solve those systems? Put a system in a form making it easy to write down a solution One form that is particularly good for this is called Row-Echelon Form: A matrix is in REF if it has the following properties: 1. If a row is not entirely zero its first nonzero entry is a 1 This entry is called a leading 1 (or a pivot) 2. Any entirely zero rows come after all nonzero rows 3. In any two nonzero rows, the leading 1 of the lower row is to the right of the leading 1 of the higher row EG: Notice a staircase pattern formed by the leading 1s

13 Solving a system in REF by back-subsitution The variables of a system in REF corresponding to columns with leading 1s are said to be bound (or leading variables). All other variables are free, and can be set equal to parameters. Solution is easily obtained by a process called back-substitution: 1. Convert augmented matrix in REF to a system of equations 2. Substitute a parameter for each free variable 3. Solve for the bound variables one per (nonzero) equation EG: We illustrate using a matrix already in REF x 1 x 2 + 2x 3 x 5 = 4 x 3 + x 4 + x 5 = 3 x 5 = 2 x 1, x 3, x 5 are bound; x 2, x 4 are free. Set x 2 = s, x 4 = t. Then x 5 = 2; x 3 = 3 x 4 x 5 = 3 t 2 = 1 t; x 1 = 4 + x 2 2x 3 + x 5 = 4 + s 2(1 t) + 2 = 4 + s + 2t Solution: (x 1, x 2, x 3, x 4, x 5 ) = (4 + s + 2t, s, t, t, 2)

14 What can go wrong? This process always solves the system... EXCEPT when... Solve for each variable starting with the rightmost and working to the left. This process always yields a solution... unless... UNLESS a leading 1 appears in the last column! If an augmented matrix in REF has a leading one in the final column there is no solution. Can you see why not? This corresponds to a system of equations in which one equation is 0 = 1 Consider the system corresponding to x + y + 2z = 3 y z = 2 0 = 1 Since 0 = 1 cannot be true the system has no solution.

15 Gaussian Elimination A method that will put any matrix into REF! A simple elimination procedure known as Gaussian elimination uses EROs to put a system into easy-to-solve REF. Perform the following 5 steps on the augmented matrix: Step 1: Locate the leftmost nonzero column of the matrix Step 2: If necessary exchange rows to bring a nonzero entry to the top position in this column Step 3: If that entry is a multiply the top row by 1 a (so that entry becomes a leading 1) Step 4: Add multiples of the top row to the lower rows so that all entries below the leading 1 become 0 Step 5: Now cover the top row. If no rows remain, stop. Otherwise, return to Step 1 and repeat the procedure on the remaining rows.

16 Reduced Row Echelon form (RREF) We can eliminate the step of back-substitution by going further with the matrix than simply REF an even stronger form A matrix is in RREF if: (1. 3.) It is in REF and If a column contains a leading 1, then every other entry in that column is 0 EG: Put this matrix in REF into RREF: R 1 R 1 2R R 1 R 1 + R

17 Solving a system in RREF in one step! If the augmented matrix of a system is in RREF then it can be solved in (essentially) one step Leading 1s are the only nonzero entries in their columns So leading (bound) variables appear in exactly one equation So you can solve for them in one step, substituting parameters for free variables! (If the last column has no leading 1) EG: x 2 x 4 = 4 x 3 + x 4 = 2 x 5 = 5 x 2 = 4 + x 4 x 3 = 2 x 4 x 5 = 5 Now x 1 and x 4 are free. Set them equal to r, s Our solution is (x 1, x 2, x 3, x 4, x 5 ) = (r, 4 + s, 2 s, s, 5)

18 Gauss-Jordan Elimination Always puts any matrix into RREF To put a matrix into RREF: (Forward phase) Use Gaussian elimination to put it into REF 6. (Backward phase) Starting with the rightmost leading 1 and working to the left, add multiples of nonzero rows to higher rows to make zero all entries above the leading 1s. EG:

19 Facts about REF and RREF In general, there are many sequences of EROs that will put a matrix in REF (and also in RREF) A given matrix can be put into more than one REF, in general. For example, ( ) ( ) ( ) R 2 R 2 R R 1 R 1 R The last two matrices are both REFs for the first one. Every matrix has, however, exactly one RREF. It follows that the positions of the leading 1s in any REF for a given matrix are the same. Columns containing leading 1s are called pivot columns.

20 Homogeneous linear systems A system of equations is homogeneous if all constant terms are zero: a 11 x 1 + a 12 x a 1n x n = 0 a 21 x 1 + a 22 x a 2n x n = 0.. a m1 x 1 + a m2 x a mn x n = 0 What can be said about the existence of a solution to such a system of equations? Every homogeneous system has at least one solution namely x 1 = x 2 = = x n = 0. This is called the trivial solution. (All other solutions are nontrivial solutions)

21 Example: homogeneous systems Trivial and nontrivial solutions For example, x 1 = x 2 = x 3 = x 4 = 0 is a solution to ( ) ( ) x 1 + 2x 2 x 4 = x 2 + x 3 = General solution (x 1, x 2, x 3, x 4 ) = (2s + t, s, s, t) includes trivial case (set s = t = 0) but also nontrivial cases (s = t = 1 gives (x 1, x 2, x 3, x 4 ) = (3, 1, 1, 1)) But in the following case: x + y = x + z = y + z = by some EROs No free variables. But trivial solution exists and is unique. Graphs of Homogeneous equations in 2D? Lines through (0, 0) In 3D? Planes through (0, 0, 0). (The origin is always common!)

22 Important facts about homogeneous systems Theorem If a homogeneous system of equations has n unknowns and the REF for its augmented matrix has r leading 1s, then the system has n r free variables... therefore a parametric solution to the system should have n r parameters If the system has m equations, then its matrix has m rows. How many leading 1s can we obtain? (At most m) What can we say about n r if m < n? (r m < n so n r > 0) Theorem A homogeneous system having more unknowns than equations has infinitely many solutions....and therefore at least one nontrivial solution!

23 Matrices ( 1.3) A matrix is a rectangular array of numbers. EG: ( ) M = Where have you encountered such arrangements of numbers? Some cases: spreadsheets... calendars... Bingo cards... our augmented matrices (of course!) The numbers in a matrix are its entries. The matrix A = a 11 a 12 a 1n a 21 a 22 a 2n..... a m1 a m2 a mn has m rows (horizontal lines) and n columns (vertical lines) The entry in the ith row and jth column (or (i, j) position) is denoted by (A) ij = a ij. What is (M) 21, above? Ans: 4

24 More elementary stuff about matrices Scalars are numerical quantities (in this case real numbers) the entries of a matrix are scalars. (Matrices are not scalars!) A matrix with m rows and n columns, a 11 a 1n A =.. a m1 a mn is said to be m n. This is the size of the matrix. If m = n it is a square matrix of order n. The main diagonal of a square matrix of order n consists of entries a 11, a 22,..., a nn. (Top left to bottom right diagonal) Two matrices A, B are said to be equal (and write A = B) if 1. They are the same size; and 2. Entries in corresponding positions are equal That is, (A) ij = (B) ij for all values of i and j

25 Matrix arithmetic Addition and subtraction of matrices The sum of matrices a 11 a 1n A =.., B = a 11 a 1n a m1 a mn b m1 b mn is obtained by adding corresponding entries: a 11 + b 11 a 1n + b 1n A + B =.. EG: ( ) ( ) = a m1 + b m1 a mn + b mn of the same size ( ) = ( 2 1 ) Similarly with subtraction of matrices. ( ) ( ) ( ) = = ( 0 3 )

26 Scalar multiplication a 11 a 1n Multiplication of a matrix A =.. a m1 a mn by a scalar c is even easier: multiply every entry of A by c. ca 11 ca 1n ca =.... ca m1 ca mn EG: 7 ( ) = ( ) = ( 7 14 ) Observe: subtraction can be performed by combining sum and scalar multiplication. A B = A + ( 1)B: ( ) ( ) ( ) = ( ) ( ) ( ) = + ( 1) =

27 Row and column vectors A 1 n matrix is called a row, or row vector. In this way all n-tuples can be considered as matrices. An m 1 matrix is called a column, or column vector. n-tuples can be represented either as rows or columns. Matrices are often partititioned into smaller parts called submatrices by inserting horizontal and vertical lines (as in the augmented matrix!). Two important partitions of a matrix: 1. Into its row vectors: ( ) ( ) r1 = = r 2 2. Into its column vectors: ( ) = = (c c 2 c 3 )

28 Matrix arithmetic Multiplication: the row-column rule One can add or subtract matrices only if their sizes match! Similarly we have a matrix multiplication that depends on matrices being compatible for this... row-column rule for multiplication If A is an m r matrix and B is an r n matrix then their product C = AB is the m n matrix whose (i, j) entry is c ij = a i1 b 1j + a i2 b 2j + + a ir b rj (multiply corresponding entries of row i of A and column j of B and add up all the results) ( ) b EG: The (1, 1) entry of the product a c d is ( e f ) 2a + c + 3e 2b + d + 3f 2a + 1c + 3e. The complete matrix: 4a + 5c + 6e 4b + 5d + 6f

29 About calculating matrix products When they re defined; what the resulting matrix looks like For the product AB to be defined: The rows of A must be the same length as the columns of B I.e., (# of columns of A) = (# of rows of B) Thus if A is m q and B is r n we must have q = r If q r then the product AB is not defined AB has same # of rows as A and same # of columns as B A helpful mnemonic: A {}}{ (m r) r inner numbers must match B {}}{ (r r n) = AB {}}{ m n and they cancel

30 Example: Which products are defined? ( ) ( ) Let A =, B =, C = Find, or give reason not defined: ( ) AB? ; AC? No (# cols of A) (# rows of C) ( ) 2 3 BA? No (# cols of B) (# rows of A); BC? CA? 2 4 ; CB? ; AA = A 2? B 2? C 2? Both no (# rows) (# cols) (not square!) Notice AB BA, BC CB etc.! ( )

31 WEEK 1: May 1-5 ACTION ITEMS Read 1.1, 1.2, 1.3 (in the text that is!) Go to our webpages and familiarize yourself their content. - Note there are two pages: a generic course page and a specific class page (for our section A01). They link to each other. - On the course page is all general information about the course, old exams, general resources provided by UM. Note particularly the Math 1300 Wiki pages designed for this course. - The class page has things specific to our class such as any handouts, lecture notes, quiz solutions, our provisional schedule of classes showing what is to be covered, and recent announcements. Link to the Math 1300 Wiki and familiarize yourself with it In particular read the entire Wiki section Systems of linear equations, Wiki pages on definition of matrices & square matrices Do all suggested homework problems from 1.1, 1.2 and the first part of 1.3 (see the course web page link to suggested homework!)

32 WEEK 1: May 1-5 SUMMARY OF TERMS DEFINED Linear equation; index/indices, double index; coefficients, constant coefficient; variables/unknowns; system of (m) linear equations; solution (of an equation or system), to solve, a solution versus the (general) solution; parameter, parametric solution; free and bound variables; consistent/inconsistent systems; augmented matrix of a system, EROs, leading 1 (pivot), pivot columns, leading variables, bound & free variables, row-echelon form (REF), reduced row-echelon form (RREF), back-substitution, Homogeneous systems, trivial & nontrivial solutions, matrix, entry, position (in a matrix), scalars, size (of a matrix), row & column indices, main diagonal, equal (matrices), sum (of matrices also addition ) and subtraction (also difference ), row/row vector, column/column vector, partitioned (matrix), submatrix, multiplication of a matrix by a scalar, and product of (two) matrices

33 WEEK 1: May 1-5 SUMMARY OF KEY CONCEPTS Line in linear refers to the graph of a 2 variable linear equation. In 3 or more variables, a the graph of a linear equation is not a line. EG, in 3 variables it is a 2-dimen- sional surface in space. However it is analogous to lines in 2 dimensions. In this case (we shall see) it is a plane. the solution to a system is also called its general solution. Most often a general solution is expressed using parameters that is, in parametric form, or it is a parametric solution. How many solutions a system might have, and the corresponding geometrical arrangement of planes and lines (in R 2 and R 3 Linear systems have: (i) no solutions (inconsistent); (ii) a unique (i.e., only one) solution (consistent); or (iii) infinitely many solutions (consistent). In 2 & 3 variables solutions lines or planes intersecting. Certain operations change a system of eq s, but not the set of solutions.

34 WEEK 1: May 1-5 SUMMARY OF KEY CONCEPTS continued The augmented matrix of a system contains all essential information about the system and can be manipulated in place of the system. Put into REF by Gaussian elimination it is easy to solve by back-substitution. A leading 1 in the last column signals no solution. Put into RREF a system can be solved in one step. RREF for any matrix is unique, but there are many REFs. There are also many sequences of EROs to put a matrix into these forms. The pivot columns and positions of leading 1s does not depend on the sequence used. Homogeneous systems always have a (trivial) solution. They might also have (infinitely many) nontrivial solutions. If m (number of equations) < n (number of unknowns) this is guaranteed; specifically there are n r free variables. Graphs of homogeneous systems are lines (in 2D) or planes (in 3D) through the origin. Difference of matrices can be written as sum and scalar mult. by 1; partitions of matrices into rows and columns

35 WEEK 1: May 1-5 SUMMARY OF METHODS LEARNED 3 operations on systems that don t affect solutions: (i) multiply an eq. by a nonzero constant; (ii) swap two eq s; (iii) add a multiple (any multiple) of one eq. to another. One can isolate a variable in one eq. and use that eq. to solve for that variable. Variables not eliminated in this way are treated as free, and set equal to parameters; solving for the other variables, the solution is given as equation of n-tuples. Converting between systems and augmented matrices, annotating EROs, Gaussian and Gauss-Jordan elimination, back-substitution, recognizing inconsistent systems by their REF. Predicting how many parameters in a solution and a nontrivial solution to a homogeneous system. Adding/subtracting matrices; product using row-column rule; knowing for which pairs of matrices sum, difference and product are (aren t) defined;