Math416_Coordinates

Math416_Coordinates - Math 416 - Abstract Linear Algebra...

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Math 416 - Abstract Linear Algebra Fall 2011, section E1 Working in coordinates In these notes, we explain the idea of working “in coordinates” or coordinate-free, and how the two are related. 1 Expressing vectors in coordinates Let V be an n -dimensional vector space. Recall that a choice of basis { v 1 ,...,v n } of V is the same data as an isomorphism ϕ : V R n , which sends the basis { v 1 ,...,v n } of V to the standard basis { e 1 ,...,e n } of R n . In other words, we have ϕ : V -→ R n v i 7→ e i v = c 1 v 1 + ... + c n v n 7→ c 1 . . . c n . This allows us to manipulate abstract vectors v = c 1 v 1 + ... + c n v n simply as lists of numbers, the coordinate vectors c 1 . . . c n R n with respect to the basis { v 1 ,...,v n } . Note that the coordinates of v V depend on the choice of basis. Notation: Write [ v ] { v i } := c 1 . . . c n R n for the coordinates of v V with respect to the basis { v 1 ,...,v n } . For shorthand notation, let us name the basis A := { v 1 ,...,v n } and then write [ v ] A for the coordinates of v with respect to the basis A . Example: Using the monomial basis { 1 ,x,x 2 } of P 2 = { a 0 + a 1 x + a 2 x 2 | a i R } , we obtain an isomorphism ϕ : P 2 -→ R 3 a 0 + a 1 x + a 2 x 2 7→ a 0 a 1 a 2 . In the notation above, we have [ a 0 + a 1 x + a 2 x 2 ] { x i } = a 0 a 1 a 2 .
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Math416_Coordinates - Math 416 - Abstract Linear Algebra...

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