Linear Algebra

112 Chapter Two. Vector Spacesholds. So, asserting that each coefficient in the lower equation is zero is the samething as asserting that c i = d i for each i, that is, that every vector is expressibleas a linear combination of the ⃗β’s in a unique way.QED1.13 Definition In a vector space with basis B the representation of ⃗v withrespect to B is the column vector of the coefficients used to express ⃗v as a linearcombination of the basis vectors:⎛ ⎞c 1c 2 Rep B (⃗v) =⎜ ⎟⎝ . ⎠c nwhere B = 〈⃗β 1 , . . . , ⃗β n 〉 and ⃗v = c 1⃗β 1 + c 2⃗β 2 + · · · + c n⃗β n . The c’s are thecoordinates of ⃗v with respect to B.Definition 1.1 requires that a basis is a sequence, that the order of thebasis elements matters, in order to make this definition possible. Without thatrequirement we couldn’t write these c i ’s in order.We will later do representations in contexts that involve more than one basis.To help keep straight which representation is with respect to which basis weshall often write the basis name as a subscript on the column vector.1.14 Example In P 3 , with respect to the basis B = 〈1, 2x, 2x 2 , 2x 3 〉, the representationof x + x 2 is⎛ ⎞0Rep B (x + x 2 1/2) = ⎜ ⎟⎝1/2⎠0(note that the coordinates are scalars, not vectors). With respect to a differentbasis D = 〈1 + x, 1 − x, x + x 2 , x + x 3 〉, the representationis different.⎛ ⎞0Rep D (x + x 2 0) = ⎜ ⎟⎝1⎠01.15 Remark This use of column notation and the term ‘coordinates’ has both adown side and an up side.The down side is that representations look like vectors from R n , which canbe confusing when the vector space we are working with is R n , especially sincewe sometimes omit the subscript base. We must then infer the intent from thecontext. For example, the phrase ‘in R 2 , where ⃗v = ( 32)’ refers to the planevector that, when in canonical position, ends at (3, 2). To find the coordinatesDB