Coordinates and Linear Transforms Coordinate Systems Defn. If B is - - PowerPoint PPT Presentation

Coordinates and Linear Transforms

Coordinate Systems

Defn. If B is a basis, then [x]B gives the coeffi- cients used to express x as a linear combination

• f vectors in B. It is called the coordinates of x

relative to B. For example, If B is

• 1
• ,
• 1

3

• and x = (5, −6), then

[x]B = (7, −2). (Check: calculate 7

• 1
• − 2
• 1

3

• .)

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Change of Coordinates

Defn. In Rn, the change-of-coordinates ma- trix PB has B as its columns. Thus x = PB [x]B where x is given in the standard basis.

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Change of Coordinates Again

Defn. For bases B and C, the change-of- coordinates matrix P

C←B

is such that [x]C = P

C←B

[x]B The columns of P

C←B

express each vector of B in terms of C. Also P

C←B

= P −1

C PB.

basDimTHREE: 4

A Linear Transform is a Matrix Transform

A linear transform T : V → W can be represented by matrix M by specifying the image of each ba- sis vector. If vectors bi form a basis B of V and the set C is a basis for W, then the columns of M are [T(bi)]C If V = W and T is identity function, then the matrix is the same as the change-of-basis ma- trix P

C←B

.

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Example

Differentiation is a linear transform from Pn to Pn−1. If, for example we take n = 3, and assume P3 and P2 have their standard bases, then dif- ferentiation is represented by    0 1 0 0 0 0 2 0 0 0 0 3    since 1 → 0, t → 1, t2 → 2t, and t3 → 3t2.

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Summary

If B is a basis, then [x]B gives the coefficients to express x as a linear combination of B. In Rn the change-of-coordinates matrix PB has B as its columns and x = PB[x]B for x in the standard basis. For two bases, the change-of-coordinates ma- trix expresses each vector of one in terms of the

• ther. Similarly, a linear transform can be rep-

resented by a matrix that specifies the image of each basis vector.

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