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Question: topic the matrix exponential...

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Topic: The matrix Exponential
The Matrix Exponential 1. Let B denote the vector space, over the real numbers, of real functions continuous on the interval [0, 1]. Define an operator T:B-B by (Ta)(t)x(t)dt, te [0, 1]. Show that for each r in B, the function Tz is also in B, and that T is a linear operator on B 2. Use the definition A to find eA when A is the diagonal matrix 00 з Hint: determine A for all n 20. 3. Compute B2 and B3, and then eB when 01 0 B 0 0 4. Use the fact that eAeB-eA+B, when A and B are square matrices and AB = BA, to compute eC when 2 1 01 C 0 2 0 0 0 1 Hint: write 2 0 0 C- A+B- 0 2 0+0 00 0 0 1 L0 0 -BA, and compute eA and e as above. check that AB
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