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340notes - Some Linear Algebra Notes An mxn linear system...

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Some Linear Algebra Notes An m x n linear system is a system of m linear equations in n unknowns x i , i = 1 , . . . , n : 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 The coefficients a ij give rise to the rectangular matrix A = ( a ij ) mxn (the first subscript is the row, the second is the column). This is a matrix with m rows and n columns: A = a 11 a 12 · · · a 1 n a 21 a 22 · · · a 2 n . . . . . . a m 1 a m 2 · · · a mn . A solution to the linear system is a sequence of numbers s 1 , s 2 , · · · , s n , which has the property that each equation is satisfied when x 1 = s 1 , x 2 = s 2 , · · · , x n = s n . If the linear system has a nonzero solution it is consistent , otherwise it is inconsistent . If the right hand side of the linear system constant 0, then it is called a homogeneous linear sys- tem. The homogeneous linear system always has the trivial solution x = 0. Two linear systems are equivalent , if they both have exactly the same solutions. Def 1.1: An m x n matrix A is a rectangular array of mn real or complex numbers arranged in m (horizontal) rows and n (vertical) columns. Def 1.2: Two m x n matrices A = ( a ij ) and B = ( b ij ) are equal , if they agree entry by entry. Def 1.3: The m x n matrices A and B are added entry by entry. Def 1.4: If A = ( a ij ) and r is a real number, then the scalar multiple of r and A is the matrix rA = ( ra ij ). If A 1 , A 2 , ..., A k are m x n matrices and c 1 , c 2 , ..., c k are real numbers, then an expression of the form c 1 A 1 + c 2 A 2 + · · · + c k A k is a linear combination of the A ’s with coefficients c 1 , c 2 , ..., c k . Def 1.5: The transpose of the m x n matrix A = ( a ij ) is the nxm matrix A T = ( a ji ). Def 1.6: The dot product or inner product of the n - vectors a = ( a i ) and b = ( b i ) is a · b = a 1 b 1 + a 2 b 2 + ... + a n b n 1
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. Example: Determine the values of x and y so that v · w = 0 and v · u = 0, where v = x 1 y , w = 2 - 2 1 , and u = 1 8 2 . Def 1.7: If A = ( a ij ) is an m x p matrix and B = ( b ij ) a p x n matrix they can be multiplied and the ij entry of the m x n matrix C = AB : c ij = ( a i * ) T · ( b * j ) . Example: Let A = 1 2 3 2 1 4 2 - 1 5 and B = - 1 2 0 4 3 5 . If possible, find AB , BA , A 2 , B 2 . Which matrix rows/columns do you have to multiply in order to get the 3 , 1 entry of the matrix AB ? Describe the first row of AB as the product of rows/columns of A and B . The linear system (see beginning) can thus be written in matrix form Ax = b . Write it out in detail. A is called the coefficient matrix of the linear system and the matrix a 11 a 12 · · · a 1 n . . . b 1 a 21 a 22 · · · a 2 n . . . b 2 . . . . . . . . . a m 1 a m 2 · · · a mn . . . b n . is called the augmented matrix of the linear system. Note: Ax = b is consistent if and only if b can be expressed as a linear combination of the columns of A with coefficients x i .
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