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Chapter 1

# Chapter 1 - Definition of Linear Equation Coefficients a...

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Definition of Linear Equation: Coefficients: a and b Variables: x 1 and x 2 Theorem: A system of linear equations has a unique solution, infinitely many solutions, or no solutions Definition of Consistent System: A system is consistent if it has a unique solution or infinitely many solutions Definition of Row Echelon Form: () may have any nonzero value. () may have any values including 0 Definition of Reduced Row Echelon Form: () may have any values including 0 The pivot columns are 2, 4, 5, 6, and 9, so the basic variables are and . The remaining variables, and , must be free variables. Theorem: If there is no “bad row,” such as 0=1, than the system must be consistent. Definition of Linear Combination: Given vector y : , is called a linear combination of v 1 , …, v p with weights c 1 , …, c p . Definition of Span : The set of all linear combinations of . Example: Let , and Is w in the Span ? Answer: Row reduce the augmented matrix . If there’s no “bad row” then it is in the span Theorem: The equation A x = b has a solution iff b is a linear combination of the columns of A. Definition of Homogeneous Linear Systems: A x = 0 Trivial solution: Theorem: The homogeneous equation has a nontrivial solution iff the equation has at least one free variable. Theorem: A homogeneous system has either a unique solution or infinitely many solutions. Linear Independence Test: The columns of matrix are linearly independent iff the equation has only the trivial solution A set of two vectors is linearly dependent if one of the vectors is a multiple of the other. The set is linearly independent iff neither vector is a multiple of the other. An indexed set of two or more vectors is linearly dependent iff at least one of the vectors is S is a linear combination of the others. A set is linearly dependent if it contains more vectors than there are entries in each vector. That is, any set

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Chapter 1 - Definition of Linear Equation Coefficients a...

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