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Unformatted text preview: CHAPTER 5 MATRICES AND THEIR USES 5.1 What is a Matrix? A system of linear algebraic equations in two vari ables might look like this: 2 x + 7 y = 3 4 x + 8 y = 11 → LINEAR because it just involves constant multiples of x and y , no x 2 , no sin( y ), etc. → ALGEBRAIC because no differentiation. It’s cool to write these systems using the following notation: • 2 7 4 8 ‚• x y ‚ = • 3 11 ‚ . Here • x y ‚ and • 3 11 ‚ are familiar  they are VEC 1 TORS. But • 2 7 4 8 ‚ is something new, called a MA TRIX. We say that the PRODUCT of • 2 7 4 8 ‚ with • x y ‚ gives you • 3 11 ‚ . Every matrix has ROWS and COLUMNS. In this case, the rows are [2 7] and [4 8] and the columns are • 2 4 ‚ , • 7 8 ‚ . We call [2 7] a ROW VECTOR and • 2 4 ‚ a column vector. We say that • 2 7 4 8 ‚ is a 2 by 2 matrix since it has two rows and two columns. You can regard [2 7] as having one row and 2 columns, etc. You can also have 3 by 3 matri ces like 1 7 9 7 8 2 4 10 12 or even 2 by 3 matrices like • 1 2 4 6 8 9 ‚ two rows, three columns. 2 A general 3 by 3 matrix can be written as a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 so a ij is the number in the ith row and jth column, Note a ij 6 = a ji usually! Engineers and physicists like to talk about “the ma trix a ij ”. Strictly speaking, they mean “the matrix with entries a ij ” but we will talk in this sloppy way too! In the same way, any column vector can be written as u 1 u 2 u 3 . 5.2 Matrix Arithmetic [a] Addition and Subtraction. Just add up or subtract the entries, as you would for a vector. 3 • 1 2 4 8 ‚ + • 7 3 6 9 ‚ = • 8 5 10 17 ‚ • 1 2 4 8 ‚ • 7 3 6 9 ‚ = • 6 1 2 1 ‚ In general, if a ij and b ij are matrices (both m by n , that is, both have m rows and n columns) then the sum is a ij + b ij and the difference is a ij b ij . [b] Multiplying By a Number. Just multiply every entry, as you would for a vector. 2 · • 1 2 4 8 ‚ = • 2 4 8 16 ‚ . The product of the number c with the matrix a ij is c · a ij . [c] Transposition. If you take a matrix and SWITCH THE FIRST ROW INTO THE FIRST COLUMN, second row into second column, and so on, the result is called the TRANSPOSE. We write • 1 2 4 8 ‚ T = • 1 4 2 8 ‚ . 4 1 7 9 6 8 2 4 10 12 T = 1 6 4 7 8 10 9 2 12 • 1 2 4 6 8 9 ‚ T = 1 6 2 8 4 9 • a 11 a 12 a 21 a 22 ‚ T = • a 11 a 21 a 12 a 22 ‚ and by looking at this example you can see a T ij = a ji → the order of the indices is reversed. Notice that ‡ ( a ij ) T · T = ( a ji ) T = a ij ( a ij + b ij ) T = a ji + b ji = a T ij + b T ij ( c a ij ) T = c a ji = c ( a ij ) T ....
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This note was uploaded on 03/29/2008 for the course MA 1506 taught by Professor Mcnines during the Spring '08 term at National University of Singapore.
 Spring '08
 McNines
 Algebra, Equations, Matrices

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