MATB24H3 Lecture Notes - Coordinate Vector, Linear Map, Linear Combination
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De nition: let v be a vector space over the eld f . B = (b1, b2, , bn) denotes an ordered basis of n vectors in v, if : b is a basis of v, the order in which the vectors appear in b is xed. Example : let b = (2, (x 2), (x 2)2, (x 2)3) be an ordered basis of the vector space p3 over the eld r. let p(x) = x3 + 5x2 x + 4. Find the coordinate vector of p relative to b. solution. Since b is a basis for p3, then there are r1, r2, r3, r4 r such that: p(x) = r1(2) + r2(x 2) + r3(x 2)2 + r4(x 2)3. = 2r1 + r2(x 2) + r3(x2 4x + 4) + r4(x3 6x2 + 12x 8) or: x3 +5x2 x+4 = r4x3 +(r3 6r4)x2 +(r2 4r3 +12r4)x+(2r1 2r2 +4r3 8r4)
