Theorem. Let B={b1,...,bn} be a basis for a vector space V. Then for each x in V, there exists unique c1,...,cn such that x=c1b1+...+cnbn.
Definition. Suppose B={b1,...,bn} is a basis for V and x is in V.
The coordinates of x relative to the basis B (or the B-coordinates of x) are the weights c1,...,cn such that c1b1+...+cnbn=x.
[x]B=⎝⎛c1...cn⎠⎞
From B-Coordinates to Standard Basis Coordinates
Ex. If b1=(10), b2=(12), B={b1,b2}, and [x]B=(−23), then find x. x=c1b1+c2b2=−2(10)+3(12)=(16).
From Standard Basis Coordinates to B-Coordinates
Ex. If b1=(21), b2=(−11), B={b1,b2}, and x=(45), then find [x]B. x(45)(45)→→→→→[x]B=c1b1+c2b2=c1(21)+c2(−11)=(21−11)(c1c2)(21−1145)(121−154)(101−35−6)(101152)(100132)=(c1c2)=(32).
Theorem. Let PB=[b1...bn]. Now x=PB[x]B.
The columns of PB are linearly independent since they form a basis, so by the Invertible Matrix Theorem, PB is invertible. This means that [x]B=PB−1x.
Let B={b1,...,bn} be a basis for V. Assigning coordinates to V gives a linear transformation from V to Rn that is one-to-one and onto. This means that V and Rn are isomorphic – they look different, but can be represented by the same mathematical object.
Ex. Let P5 be the set of all polynomials of degree five or less.
{1,t,t2,t3,t4,t5} is a basis for P5. This means any polynomial of degree five or less P takes the form P(x)=a5x5+a4x4+a3x3+a2x2+a1x1+a0.
If T is the transformation that assigns coordinates from P5 to R6, then T(P(x))=⎝⎜⎜⎜⎜⎜⎜⎛a5a4a3a2a1a0⎠⎟⎟⎟⎟⎟⎟⎞
Thus P5 is isomorphic to R6. This means we can perform computations like adding two polynomials in P5 by adding their corresponding vectors in R6.
Ex. B={sint, cost} is the basis of a subspace inside of C∞(R). (sint and cost are linearly independent functions.) Since B only contains two vectors, functions within this subspace can be represented as vectors in R2.