Lect6 - Chapter 1 Matrices and Systems of Equations Math1111 Matrix Algebra Linear Combination Definition n Definition Let a1 a2 an be vectors in

# Lect6 - Chapter 1 Matrices and Systems of Equations...

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Chapter 1. Matrices and Systems of Equations Math1111 Matrix Algebra Linear Combination - Definition Definition Let a 1 , a 2 , ··· , a n be vectors in n , and c 1 , c 2 , ··· , c n be scalars. Then a sum of the form c 1 a 1 + c 2 a 2 + ··· + c n a n is called a linear combination of the vectors a 1 , a 2 , ··· , a n .
Chapter 1. Matrices and Systems of Equations Math1111 Matrix Algebra Linear Combination - Definition Definition Let a 1 , a 2 , ··· , a n be vectors in n , and c 1 , c 2 , ··· , c n be scalars. Then a sum of the form c 1 a 1 + c 2 a 2 + ··· + c n a n is called a linear combination of the vectors a 1 , a 2 , ··· , a n . Example . Is 1 2 a linear combination of 1 1 , 2 1 , 1 3 ? Is 1 1 2 a linear combination of 1 1 1 , 0 1 0 , 1 0 1 ?
Chapter 1. Matrices and Systems of Equations Math1111 Matrix Algebra Consistency
Chapter 1. Matrices and Systems of Equations Math1111 Matrix Algebra Consistency Theorem A x = b is consistent if and only if b is a linear combination of column vectors of A . Proof. “if”: Write A = ( a 1 a 2 ··· a n ) . Suppose b is a linear combination of a 1 , a 2 , ··· , a n . It means that we have some scalars α 1 , α 2 , ··· , α n such that α 1 a 1 + α 2 a 2 + ··· + α n a n = b .
Chapter 1. Matrices and Systems of Equations Math1111 Matrix Algebra Consistency Theorem A x = b is consistent if and only if b is a linear combination of column vectors of A . Proof. “if”: Write A = ( a 1 a 2 ··· a n ) . Suppose b is a linear combination of a 1 , a 2 , ··· , a n . It means that we have some scalars α 1 , α 2 , ··· , α n such that α 1 a 1 + α 2 a 2 + ··· + α n a n = b . Take x = α 1 α 2 . . . α n , then A x = b . i.e. The equation A x = b has solution.
Chapter 1. Matrices and Systems of Equations Math1111 Matrix Algebra Consistency (Cont’d) Proof. “only if”: Let a 1 a 2 . . . a n be a solution of A x = b . i.e. A a 1 a 2 . . . a n = b .
Chapter 1. Matrices and Systems of Equations Math1111 Matrix Algebra Consistency (Cont’d) Proof. “only if”: Let a 1 a 2 . . . a n be a solution of A x = b . i.e.