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MA242Final

MA242Final - Linear Algebra Final Exam-consistent=rightmost...

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Linear Algebra Final Exam - consistent =rightmost col of augmatrix is NOT a pivot col unique =no free variables -Ax=b has a sol for all b→each b in R is a lin combo of cols of A→cols of A span R→A has a pivot pos in every row - homogeneous (Ax=0)has nontrivial sol if/only if the eq has at least 1 free var - linearly independent =(Ax=0 has only trivial sol→no free variables)(not multiples of e/o) - linearly dependent =(at least one vector is a multiple of the other)(#col>#rows)(contains 0vector) - onto(surjective) =each b in R is the image at least one x (colA span R) - one-to-one(injective) =each b in R is the image of at least one x(only trivial sol→linearly ind) -if A is invertible →Ax=b has unique sol(lin ind) → if A is invertible↔injective↔surjective↔bijective →if detA≠0→A is invertible →(det(A))(det(B))=det(AB) - Area =|detA| - Volume =|detB| - area of T(s) =|detA|(area of S) - vector space =nonempty set of vectors defined by addition and scalar multiplication - subspace
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