Lecture 17 COORDINATES IN VECTOR SPACES MATH 20580

Lecture 17 COORDINATES IN VECTOR SPACES MATH 20580 Linear Algebra & Differential Equations, Notre Dame. Instructor: Dmitriy Voloshyn

Class Participation from 03/12

Coordinates for polynomials Once we chose a basis for a vector space, we can assign to each object its columns of coordinates and forget about the nature of the objects of the vector space (if we only use linear algebra). Columns completely encode all the relevant “linear” information. For example, adding two polynomials is the same as adding the columns, and similarly for scaling. All “linear questions” about objects of vector spaces can be then translated to “linear questions” about the columns. Example.

Coordinates for matrices (the convention on the order is as in the textbook) Similarly, adding two matrices is the same as adding the columns assigned to them. The matrices from the standard matrices are: This interpretation is used for finding coordinates in any other bases.

Coordinates for matrices: example Solution. We learned one way a week ago. Another way: assign columns to the matrices. (read off coordinates from up to down, from left to right) Set up a linear system (given in terms of its augmented matrix) Solve the system (e. g. , find RREF)

Coordinates for polynomials: example Solution. Assign columns to all polynomials: Concatenate the columns to form an augmented matrix: Row reduce

Change of basis in polynomial spaces Solution. Assign coordinate columns to all entities: Once we have columns, everything is the same as on week 5 (coordinates of vectors). Can use the formula

Change of basis in polynomial spaces continued 2. Apply the Gauss-Jordan elimination algorithm to find the RREF of this matrix:

Why does it work?

Back to the problem Or

Change of basis in spaces of matrices RREF

Change of basis in spaces of matrices continued We found before This means