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The following example illustrates how Theorem 6.1.1 and the defining properties of inner products can be used to perform
algebraic computations with inner products. As you read through the example, you will find it instructive to justify the steps.
EXAMPLE 11 Calculating with Inner Products
Since Theorem 6.1.1 is a general result, it is guaranteed to hold for all real inner product spaces. This is the real power of the
axiomatic development of vector spaces and inner products—a single theorem proves a multitude of results at once. For example,
we are guaranteed without any further proof that the five properties given in Theorem 6.1.1 are true for the inner product on
generated by any matrix A [Formula 3]. For example, let us check part (b) of Theorem 6.1.1 for this inner product:
The reader will find it instructive to check the remaining parts of Theorem 6.1.1 for this inner product.
Exercise Set 6.1
Click here for Just Ask!
Let be the Euclidean inner product on , and let ,, , and . Verify that
1.
(a)
(b)
(c)
(d)
(e)
