§4.4 Coordinate Systems in General, People Are More Comfortable Working with Rn and Its Subspaces Than with Other Types of Vect

§4.4 Coordinate Systems

In general, people are more comfortable working with Rn and its subspaces than with other types of vector spaces and subspaces. The goal in this section is to deﬁne an amazing correspondence between Rn and other vector spaces that allows us to do calculations in Rn to learn about more general vector spaces. Theorem 7 The Unique Representation Theorem

Let B = {b1, b2,..., bn} be a basis for a vector space V . Then for each vector x ∈ V, there exists a unique set of scalars c1, . . . , cn such that x = c1b1 + ··· + cnbn.

dfn: Suppose B = {b1, b2,..., bn} is a basis for a vector space V and x ∈ V . The coordinates of x relative to the basis B, or the B-coordinates of x, are the weights c1, . . . , cn such that

x = c1b1 + ··· + cnbn.   c1  .  In this case, the vector [x]B =  .  is called the coordinate vector of x relative to B, or the B-coordinate cn n vector of x. Notice that [x]B is a vector in R .

Coordinates in Rn

  x1 x2  When we are working in n and we write x =  , we have written the coordinates of our vector x with respect to the R  .   .  xn n standard basis {e1, e2,..., en} of R . We will now discuss how to compute a change of coordinates from an arbitrary basis for Rn to the standard basis and visa-versa.

  c1  c2  Given a basis B = {b , b ,..., b } for n, deﬁne the matrix P = [ b b ··· b ] . Then if [x] =  , the 1 2 n R B 1 2 n B  .   .  cn vector equation x = c1b1 + ··· + cnbn is equivalent to the matrix equation

x = PB[x]B.

This formula allows us to ﬁnd x from its coordinates in the B basis. We call PB the change of coordinates matrix from B to the standard basis in Rn. −1 Since PB is a square matrix with linearly independent columns, PB exists by the Invertible Matrix Theorem, so we can conclude that −1 [x]B = PB x. −1 n Therefore PB is the change of coordinates matrix from the standard basis in R to the basis B.

Isomorphisms and Coordinate Mappings dfn: Let T : V → W be a linear transformation. Then T is an isomorphism if T is one-to-one and T is onto. (Recall T is one-to-one: if, for all u and v in V , if T (u) = T (v), then u = v (equivalently, since T is linear, x = 0 is the only solution to T (x) = 0). T is onto: if for every w ∈ W there is a v ∈ V with T (v) = w.)

Choosing an ordered basis B = {b1,..., bn} for a vector space V introduces a coordinate system in V . That is, the coordinate mapping x 7→ [x]B (sometimes written C(x) = [x]B) connects the possibly unfamiliar space V with the familiar space Rn. Points in V can now be identiﬁed by their coordinates in Rn, and every vector-space calculation in V is accurately reproduced in Rn (and vice versa–see the Coordinate Map Theorem on the next page!). Note that V is not Rn but V does look like Rn as a vector space. (over) −→ Theorem 8 Let B = {b1, b2,..., bn} be a basis for a vector space V . Then the coordinate mapping x 7→ [x]B is an isomorphism from V to Rn (i.e., linear, one-to-one, and onto).

Remarks: • Linearity of the coordinate mapping means for all choices of weights and vectors in V , that

[c1u1 + ··· + cpup]B = c1[u1]B + ··· + cp[up]B.

• Two isomorphic spaces (that is, two spaces linked by a one-to-one, onto linear transformation) may look like entirely diﬀerent spaces, but as vector spaces they act the same!! This is evidenced by the fact that every vector 3 space calculation in one space is accurately reproduced in the other, as in the parallel universes of P2 and R : 3 3 Example: The parallel universes of P2 and R : P2 is isomorphic to R by the coordinate map p 7→ [p]E where 2 E = {1, t, t } is the standard basis in P2:

3 P2 R

 a  p(t) = a + bt + ct2  b  c

 −1   2   1  (−1 + 2t − 3t2) + (2 + 3t + 5t2) 2 + 3 = 5 = 1 + 5t + 2t2       −3 5 2

 0   0  4(5t − 6t2) = 20t − 24t2 4  5  =  20  −6 −24

Coordinate Map Theorem Let B = {b1, b2,..., bn} be a basis for a vector space V . Let S = {v1,..., vp} be an indexed set of vectors in V . Then,

i. {v1,..., vp} is independent in V if and only if the set of coordinate vectors {[v1]B,..., [vp]B} is independent in Rn. n ii. {v1,..., vp} spans V if and only if the set of coordinate vectors {[v1]B,..., [vp]B} spans R .

iii. the vector w ∈ V is a linear combination of {v1,..., vp} if and only if the coordinate vector [w]B is a linear combination of {[v1]B,..., [vp]B}.

Moral: If B = {b1, b2,..., bn} is a basis for V and S = {v1,..., vp} an indexed set of vectors in V , then

n • To check if S = {v1,..., vp} is independent in V , check if {[v1]B,..., [vp]B} is independent in R . n • To check if S = {v1,..., vp} spans V , check if {[v1]B,..., [vp]B} span R .

• To check if w ∈ V is a linear combination of S = {v1,..., vp}, check if [w]B is a linear combination of {[v1]B,..., [vp]B}.