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Orthogonal Matrices
The proof of the equivalence is nearly identical to the proof of Theorem 2 from the previous section and so we’ll leave it to you to fill in the details. Since it is much easier to verify that the columns/rows of a matrix are orthonormal than it is to...
Orthonormal Basis
and yes it does require an inner product space to construct. However, before we do that we’re going to need to get some preliminary topics out of the way first. We’ll first need to get a set of definitions out of way. Definition 1 Suppose that S is...
Convolution Integral
take the inverse transforms of a product of transforms. Fact Let’s work a quick example to see how this can be used. Example 1 Use a convolution integral to find the inverse transform of the following transform. Solution First note that we could use #11 fro...
Periodic Functions and Orthogonal Functions
definition of orthogonal functions out of the way. Definition 1. Two non-zero functions, and , are said to be orthogonal on if, 2. A set of non-zero functions, , is said to be mutually orthogonal on (or just an orthogonal set if we’re be...
Inverse Laplace Transforms
entry in the table has a particular denominator, then the numerators of each will be different, so go up to the numerator and see which one you’ve got. If you need to correct the numerator to get it into the correct form and then take the inverse transform. So, wi...
Nonconstant Coefficient IVP�s
of our solution. So, the transform of our solution, as well as the solution is, I’ll leave it to you to verify that this is in fact a solution if you’d like to. Now, not all nonconstant differential equations need to use (1). So, let’s take a...
Solving IVP�s with Laplace Transforms
an IVP we will need initial conditions at t = 0. While Laplace transforms are particularly useful for nonhomogeneous differential equations which have Heaviside functions in the forcing function we’ll start off with a couple of fairly simple problems to illustrate how the proces...
Step Functions
would be to use the formula to break it up into cosines and sines with arguments of t-5 which will be shifted as we expect. There is an easier way to do this one however. From our table of Laplace transforms we have #16 and using that we can see that if This wi...
Equations of Planes
Recall from the Dot Product section that two orthogonal vectors will have a dot product of zero. In other words, This is called the vector equation of the plane. A slightly more useful form of the equations is as follows. Start with the first form of the vector equation and write d...
IVP�s with Step Functions
the forcing function. This is where Laplace transform really starts to come into its own as a solution method. To work these problems we’ll just need to remember the following two formulas, In other words, we will always need to remember that in order t...
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