algebra: definitions and theorem

May 29th, 2015
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Vector Space: has zero element, closed under addition and subtraction (which are commutative & associative), scalar multiplication is associative I guess, etc. i.e. vector spaces include set of polynomials of degree n, R^n, M(m,n) (the set

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MAT 204: Definitions and Theorems (Midterm Prep) Vector Space: has zero element, closed under addition and subtraction (which are commutative & associative), scalar multiplication is associative I guess, etc. i.e. vector spaces include set of polynomials of degree n, , M(m,n) (the set of m by n matrices), etc?** do not include things not closed under addition and subtraction, like positive values of R^n or whateverGaussian Elimination and such: Basically write out the augmented matrix (coefficients + constants) Echelon form: the first nonzero entry in any nonzero row occurs to right of first entry in the row DIRECTLY above itAll zero rows grouped together at bottomRow-Reduced Echelon form: A is in echelon formAll pivot entries are 1All entries above pivots are 0Theorem: (More Unknowns):If there are more unknowns than there are equations (rank of m by n matrix is m, m

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