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midterm_1_topics - MA 265 REVIEW FOR MIDTERM#1 GOINS 1...

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MA 265 GOINS REVIEW FOR MIDTERM #1 § 1: Linear Equations and Matrices 1.1: Systems of Linear Equations a system of m linear equations in n unknowns is in the form a 11 x 1 + a 12 x 2 + · · · + a 1 n x n = b 1 a 21 x 1 + a 22 x 2 + · · · + a 2 n x n = b 2 . . . . . . . . . . . . a m 1 x 1 + a m 2 x 2 + · · · + a mn x n = b m If the system has a solution, we say it is consistent . Otherwise, we say it is inconsistent . If b 1 = b 2 = · · · = b m = 0, when we call this system a homogeneous system . Otherwise, we call it an inhomogeneous system . A homogeneous system is always consistent because one has the trivial solution x 1 = x 2 = · · · = x n = 0. A second system of equations c 11 x 1 + c 12 x 2 + · · · + c 1 n x n = d 1 c 21 x 1 + c 22 x 2 + · · · + c 2 n x n = d 2 . . . . . . . . . . . . c r 1 x 1 + c r 2 x 2 + · · · + c rn x n = d r is equivalent to the first if they have exactly the same solutions. 1.2: Matrices An m × n matrix is an array with m columns and n rows: A = bracketleftbig a ij bracketrightbig = a 11 a 12 . . . a 1 n a 21 a 22 . . . a 2 n . . . . . . . . . . . . a m 1 a m 2 . . . a mn . The i th row and j th columns of A are row i ( A ) = bracketleftbig a i 1 a i 2 · · · a in bracketrightbig and col j ( A ) = a 1 j a 2 j . . . a mj , respectively. An n -vector is an n × 1 matrix: x = x 1 x 2 . . . x n . If A is an n × m matrix and B is a q × p matrix, then we say A = B if n = q , m = p , and a ij = b ij . A a square matrix if m = n . The numbers a 11 , a 22 , . . . a nn form the main diagonal . Matrix addition (the sum ) and subtraction (the difference ) is defined componentwise for matrices of the same size: A = bracketleftbig a ij bracketrightbig , B = bracketleftbig b ij bracketrightbig = A ± B = bracketleftbig a ij ± b ij bracketrightbig . 1
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2 MA 265 MIDTERM #1 REVIEW Scalar multiplication is the product of an m × n matrix A and a number r : A = bracketleftbig a ij bracketrightbig = r A = bracketleftbig r a ij bracketrightbig . A linear combination of a collection A 1 , A 2 , . . . , A k of m × n matrices and a collection c 1 , c 2 , . . . , c k of numbers is the m × n matrix c 1 A 1 + c 2 A 2 + · · · + c k A k . Summation notation is short-hand for expressing such a sum: k summationdisplay i =1 a i = a 1 + a 2 + · · · + a k . The transpose of an m × n matrix A is that n × m matrix A T formed by interchanging the rows and columns of A : A = a 11 a 12 . . . a 1 n a 21 a 22 . . . a 2 n . . . . . . . . . . . . a m 1 a m 2 . . . a mn = A T = a 11 a 21 . . . a n 1 a 12 a 22 . . . a n 2 . . . . . . . . . . . . a 1 m a 2 m . . . a nm . 1.3: Matrix Multiplication The dot product (or inner product ) of two n -vectors is a number: a = a 1 a 2 . . . a n , b = b 1 b 2 . . . b n = a · b = n summationdisplay k =1 a k b k = a 1 b 1 + a 2 b 2 + · · · + a n b n .
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