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We are going to study matrices and vectors. First, let's load up the necessary package for doing linear algebra:

with(LinearAlgebra); Construct a Matrix using shortcut notation C := < < 1 | 2 | 3 > , < 4 | 5 | 6 >, <7 | 8 | 9> >;

More matrix constructions with algebra: A := Matrix([[2, 5, -3], [1, -1, 3], [4, 0 , 2]]); Now construct the matrix B=[[6,0,3],[0,1,0],[2,3,0]] using the GUI tools on the left side of this sheet. 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 Performing calculations in a casual usage scenario accepts optional arguments in any order. MatrixAdd(A, B); 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 Now construct a few vectors: a := Vector([1,2,3]); O.K., now do it with the GUI tools on the left to construct (0,-1,-2):LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn PkkiYkc2Ii0mSSdWZWN0b3JHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0knY29sdW1uR0YkNiMvSSQlaWRHRiQiKiFbViVRIg== PkkiY0c2IiwmSSJhR0YkIiIiSSJiR0YkRic= LUk1TWF0cml4VmVjdG9yTXVsdGlwbHlHNiI2JEkiQUdGJEkiYkdGJA== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn LUk1TWF0cml4TWF0cml4TXVsdGlwbHlHNiI2JEkiQUdGJEkiQkdGJA== Example 2.18 of text (Toothpaste marketing) Let's figure out what happens if we run an initial stage to 5, 10 and 20 iterations PkkiQUc2IiwkKiYjIiIiIiM1RigtSSdNYXRyaXhHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2IzckNyQiIigiIiU3JCIiJCIiJ0YoRig= PkkjeDBHNiItSSdWZWN0b3JHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2IzckIiIiIiIh LUk1TWF0cml4VmVjdG9yTXVsdGlwbHlHNiI2JC1JLE1hdHJpeFBvd2VyR0YkNiRJIkFHRiQiIiZJI3gwR0Yk Example 2.20 of text (Biology example) -- use the GUI here to get [[0.2, 0, 0.25],[0.6, 0.3, 0], [0, 0.6, 0.8]]: PkkiQUc2Ii1JJ01hdHJpeEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjL0kkJWlkR0YkIipbJT5oOA== Now let's study behavior over time PkkjeDBHNiItSSQ8LD5HSShfc3lzbGliR0YkNiUiJCsiIiMhKSIjZw==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn MatrixVectorMultiply(MatrixPower(A, 5), x0); Example 2.21 of text (graph example). Let's find the number of walks of length at most three in the graph. Here the adjacency matrix is given by A := Matrix([[0,1,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]); LChJIkFHNiIiIiItSSxNYXRyaXhQb3dlckdGJDYkRiMiIiNGJS1GJzYkRiMiIiRGJQ== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn