1300 Linear Algebra and Vector Geometry Week 2: Jan , Gauss-Jordan, homogeneous matrices, intro matrix arithmetic

Transcription

1 1300 Linear Algebra and Vector Geometry Week 2: Jan , Gauss-Jordan, homogeneous matrices, intro matrix arithmetic R. Craigen Office: MH Winter 2019

2 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 with augmented matrix x + y + 2z 3 y z Since 0 1 cannot be true the system has no solution.

3 Gaussian Elimination A method that will put any matrix into REF! A simple procedure known as Gaussian elimination uses EROs to put a system into easy-to-solve REF; and it always works! 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: Omitting the top row from consideration, if no nonzero rows remain, then stop. Otherwise, return to Step 1 and repeat the procedure on the remaining rows.

4 Examples of Gaussian Elimination

5 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 2R 2 R R 1 + R 3 R

6 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 the leading columns So each leading (bound) variable appears 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 2x 4 4 x 3 + 2x 4 2 x 5 5 x x 4 x 3 2 x 4 x 5 5 Now x 1 and x 4 are free. Set them equal to parameters r, s Our solution is (x 1, x 2, x 3, x 4, x 5 ) (r, 4 + 2s, 2 s, s, 5)

7 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:

8 Facts about REF and RREF In general, there are many sequences of EROs that will put a matrix into REF (and also into RREF) A given matrix can be put into more than one REF, in general. For example, ( ) R 2 R 2 R 1 ( ) R 1 R 1 R 2 ( 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 or just leading columns )

9 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)

10 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 x 2 + x 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!)

11 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!

12 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

13 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

14 Matrix arithmetic Addition and subtraction of matrices The sum of matrices a 11 a 1n A.., B b 11 b 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 )

15 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)

16 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: ( ) ( ) r r 2 2. Into its column vectors: ( ) (c c 2 c 3 )

17 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

18 Week 2: Jan Summary ACTION ITEMS Read 1.2, 1.3 Do associated homework questions. On the webpage watch for practice materials connected to current learning. Do worksheets 0, 1 and 2 available there (future ones will probably NOT appear on the web page but will be distributed in lab) The web page has things specific to our class such as any handouts, lecture notes, quiz solutions, our provisional schedule of classes (eventually I hope to post something there!) showing what is to be covered, and recent announcements. Do all suggested homework problems from 1.2, 1.3 up to questions about matrix multiplication See the web page link to suggested homework

19 Week 2: Jan Summary SUMMARY OF TERMS DEFINED reduced row-echelon form (RREF), Homogeneous systems, trivial & nontrivial solutions, matrix, entry, position (in a matrix), scalars, size (of a matrix), row & column indices, main diagonal, equal (matrices), sum (also addition ) and subtraction (also difference ) of matrices, row/row vector, column/column vector, partitioned (matrix), submatrix, multiplication of a matrix by a scalar, and product of (two) matrices (covered in Week 3 in Section A01)

20 Week 2: Jan Summary SUMMARY OF KEY CONCEPTS 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; but Gaussian elimination and Gauss-Jordan elimination, respectfully, are guaranteed to do so. The pivot columns and positions of leading 1s do not depend on the sequence of EROs 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) nontrivial solutions exist; specifically r leading 1s lead to n r free variables. Graphs of homogeneous systems are lines (in 2D) or planes (in 3D) through the origin. Matrices are rectangular arrays of numbers, and there is an arithmetic of matrices to be learned. Equality, addition, difference are entry-wise. Difference of matrices can be written as a sum together with a scalar mult. by 1; partitions of matrices into rows and columns

21 Week 2: Jan Summary SUMMARY OF METHODS LEARNED Gaussian and Gauss-Jordan elimination, recognizing inconsistent systems by their REF. Predicting how many parameters in a solution and a nontrivial solution to a homogeneous system. Counting parameters by counting leading 1s; Adding/subtracting matrices; product using row-column rule (Section A01 in week 3); knowing for which pairs of matrices sum, difference and product are (aren t) defined