MA242Final

MA242Final - Linear Algebra Final Exam - consistent...

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Unformatted text preview: 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 beach b in R is a lin combo of cols of Acols of A span RA 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 solno 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 sollinearly ind)-if A is invertible Ax=b has unique sol(lin ind) if A is invertibleinjectivesurjectivebijective if detA0A 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 has 0vectorfor each u&v u+v is in Hfor each u cu is in H...
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MA242Final - Linear Algebra Final Exam - consistent...

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