MATH 133 Final Exam 2023 With Complete solution linear equation - an equation whose graph is a straight line; all variables have the
power 1 and there is no multiplication between them: ax + by = c
system of linear equations - a finite, non-empty set of linear equations unique solution - when there is only 1 intersecting point; the system is consistent no solution - the lines are parallel or distinct; the system is inconsistent infinitely many solutions - the lines are identical; the system is consistent; the set of solutions is said to be given in parametric form and is called the general solution to the system augmented matrix - a coefficient matrix with an extra column containing the constant terms (the constant matrix) elementary operations - I) interchange two equations II) multiply one equation by a nonzero number III) add a multiple of one equation to a different equation reversed elementary row operations - I) interchanging the rows again II) multiplying the row by 1/k III) adding -k times row p to row q (p cannot equal q) matrix - a rectangular array of numbers row echelon form - 1) all zero rows are at the bottom 2) the first non-zero entry in each nonzero row is a 1 (the leading 1) 3) each leading 1 is to the right of the leading 1's in the rows above it reduced row echelon form - row echelon form where each leading 1 is the only nonzero entry in its column rank of a matrix - the number of leading 1s in the REF of the matrix gaussian algorithm - 1) move row with nonzero first entry, a, to the top 2) multiply by 1/a to create leading one 3) subtract multiples of that row from rows below it, make each entry below the leading 1 zero. REPEAT gaussian elimination - carry the augmented matrix to RREF. parametrize and use equations to solve for leading variables in terms of parameters
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