Transcription of MATH 304 Linear Algebra
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MATH 304 Linear Algebra
math 304 Linear AlgebraLecture 14:Basis and of and a vector space. A linearlyindependent spanning set forVis called vector spaceVhas a basis. IfVhas a finite basis, then all bases forVare finite andhave the same number of elements (called thedimensionofV). (1,0,0, .. ,0,0),e2= (0,1,0, .. ,0,0),.. ,en= (0,0,0, .. ,0,1)form a basis forRn(calledstandard) since(x1,x2, .. ,xn) =x1e1+x2e2+ + and coordinatesIf{v1,v2, .. ,vn}is a basis for a vector spaceV,then any vectorv Vhas a unique representationv=x1v1+x2v2+ +xnvn,wherexi R. The coefficientsx1,x2, .. ,xnarecalled thecoordinatesofvwith respect to theordered basisv1,v2, .. , mappingvectorv7 its coordinates(x1,x2, .. ,xn)is a one-to-one correspondence correspondence respects Linear operations inVand Coordinates of a vectorv= (x1,x2, .. ,xn) Rnrelative to the standardbasise1= (1,0, .. ,0,0),e2= (0,1.)
MATH 304 Linear Algebra Lecture 14: Basis and coordinates. Change of basis. Linear transformations. Basis and dimension Definition. Let V be a vector space. A linearly independent spanning set for V is called a basis. Theorem Any vector space V has a basis. If V ... 0 1 1 . The ...
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