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 of vectors in B . It is called the coordinates of x relative to B . �� � � �� 1 1 For example, If B is and x = (5 , − 6) , then , 0 3 � � � � 1 1 [ x ] B = (7 , − 2) . (Check: calculate 7 .) − 2 0 3 basDimTHREE: 2

Change of Coordinates In R n , the change-of-coordinates ma- Defn. trix P B has B as its columns. Thus x = P B [ x ] B where x is given in the standard basis. basDimTHREE: 3

Change of Coordinates Again Defn. For bases B and C , the change-of- coordinates matrix is such that P C ← B [ x ] C = [ x ] B P C ← B The columns of express each vector of B in P C ← B = P − 1 terms of C . Also C P B . P C ← B 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 b i form a basis B of V and the set C is a basis for W , then the columns of M are [ T ( b i )] 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 basDimTHREE: 5

Example Differentiation is a linear transform from P n to P n − 1 . If, for example we take n = 3 , and assume P 3 and P 2 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 , t 2 �→ 2 t , and t 3 �→ 3 t 2 . basDimTHREE: 6

Summary If B is a basis, then [ x ] B gives the coefficients to express x as a linear combination of B . In R n the change-of-coordinates matrix P B has B as its columns and x = P B [ 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 other. Similarly, a linear transform can be rep- resented by a matrix that specifies the image of each basis vector. basDimTHREE: 7