GLOSSARY: A DICTIONARY FOR LINEAR ALGEBRA
Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node
i to node j; otherwise aij = 0. A = AT for an undirected graph.
Affine transformation T (v ) = Av + v 0 = linear transformati
Part I Problems
Problem 1: Let be a given complex number. From the denition of the Laplace trans
form, nd L ezt and also its region of convergence.
Problem 2: By using the table of formulas, nd:
(a) L et sin 3t
(b) L e2t (t2 3t + 2) .
Problem 3: Find L
Part II Problems
Problem 1: [Laplace transform] (a) Suppose that F (s) is the Laplace transform of f (t),
and let a > 0. Find a formula for the Laplace transform of g(t) = f ( at) in terms of F (s),
by using the integral denition and making a change of va
Computing the Laplace Transform
Quiz: What is L(1 + t)2 )?
Think about your answer and then look at the choices.
18.03SC Differential Equations
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18.03SC Practice Problems 27
Rules for the Laplace transform
f (t)est dt
L[ f (t)] = F (s) =
L[ a f (t) + b g(t)] = a F (s) + b G (s).
L 1 :
F (s) essentially determines f (t) for t > 0.
L[ert f (t)] =