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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...
Dot Product & Cross Product
which we’ll denote for now, will be parallel to the vector a and the other, denoted for now, will be orthogonal to a. See the image below to see some examples of kind of decomposition. From these figures we can see how to actually construct the two...
Inner Product Spaces
we are working with an inner product space. The following definition gives one such subspace. Definition 5 Suppose that W is a subspace of an inner product space V. We say that a vector u from V is orthogonal to W if it is orthogonal to every vect...
Vector Spaces
segment in or . Nor does a vector have to represent the vectors we looked at in . As we’ll soon see a vector can be a matrix or a function and that’s only a couple of possibilities for vectors. With all that said a good many of our examples will be examples from since...
-Decomposition
So, we can factor A as a product of Q and R and Q has the correct form. Now all that we need to do is to show that R is an invertible upper triangular matrix and we’ll be done. First, from the Gram-Schmidt process we know that is orthogonal to , , &...
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