Homework: Read 5.1 and 5.2 carefully. Do and turn in 5.1: 3 ,4,8,9,11,17 and 5.2: 2(a,b,e,f), 3(a,b,d),7,9,13.

Proposed topics Diagonizability. Last Friday we introduced to the notions of an eigen-value and eigen-vector. This week we will apply these notions to choosing a good basis in which to veiw a linear transformation T. We should think of a very nice basis $\beta$ as being one where $[T]_{\beta}$ is diagonal. Sometimes we can find such a basis sometimes we can't. This depends on several factors: which feild we are working in, how many eigenvalues there are in this feild, and how many eigen-vectors we can find for each eigenvalue. We will explore these possibilities. (Warning: many of the corollaries stated in 5.2 and 5.3 we will simply deduce every time we use them (since they are very easily deduced form the threorem), however you should read them all very carefully to get a feel for what these wonderous theorem are giving us.)

Some Maple Code: Here the maple code form last Friday and this Monday:

with(linalg);

T2 := matrix([[3/5, 1/2, 1/10], [1/5, 2/5, 1/5], [1/5, 1/10, 7/10]]); Change := matrix([[8, 1, 4], [5, 0, -3], [7, -1, -1]]);

Identity3 := diag(1, 1,1); Augmented := augment(Change,Identity3);

S1 := gaussjord(Augmented);

Changeback := transpose(matrix([col(S1,4..6)]));

NewT2 := multiply(Changeback,T2,Change);

Lim := diag(1,0,0);

Prob1 := matrix([[1],[0],[0]]);

Stationary := multiply(Change,Lim,Changeback,Prob1);

Observe := multiply(T2,Stationary);

Another1 := multiply(Change,Prob1);

Standard2 := matrix([[0],[1],[0]]);

Another2 := multiply(Change,Standard2);

Standard3 := matrix([[0],[0],[1]]); Another3 := multiply(Change,Standard3);

multiply(T2,Another1);

multiply(T2,Another2);

multiply(T2,Another3);