Note: I am not an official teacher or GSI for any of the courses listed on this site unless otherwise stated. Any problems and notes are created from my own observations and may not accurately reflect the course's content.

Linear Transformations #1

Three tea shops, shop A, shop B, and shop C, control the entirety of the Berkeley boba market. At a time where t = 0, the shares of the market are: A, 0.4; B, 0.2; C, 0.4. During the first year, shop A retained 85% of its customers, lost 5% to shop B, and lost 10% to shop C. Shop B retained 75%, and lost 15% to shop A, 10% to shop C. Shop C retained 90% and lost 5% to A, 5% to B. Assume that this trend does not change and the Berkeley boba market does not expand or contract.

Problem A

Write the transition matrix, matrix A, for the first year.

Solution to Problem A

Problem B

Write a general form for the next year, x_t+1 in terms of the transition matrix, A, and the current state, x_t .

Solution to Problem B

Problem C

Find the market share of each boba shop after the first year using the transiton matrix, A, and the original market share values, x₀

Solution to Problem C

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