Let . Prove that for any non-zero vector .
Prove that with is an inner product space.
Using , find the norms and angle between the vectors and .
Determine which of these bilinear forms are inner products.
Let , the 2-dimensional real vector space; define for all . Here is the real part of , and is the complex conjugate of .
Let . Define .
For each of the following subspaces of find an orthogonal basis.
.
.
The row space of .
The orthogonal complement of .
Let , and define the bilinear form
for . This defines an inner product. Apply the Gram-Schmidt process to the basis to produce an orthogonal basis for .
In an inner product space, prove that an orthogonal sequence of non-zero vectors is always linearly independent.
Let . A student is asked to prove that , and he writes the following:
[Student box]
If , and , then , by definition of .
But we know , and therefore
Hence .
[End of Student box]
What has the student done wrong, and how might he get full marks?
A student is asked to prove Theorem 3.11, and she writes the following:
[Student box]
Assume that is symmetric.
Notice that for any standard basis vectors we have that
So implies that .
Therefore is a symmetric matrix.
[End of Student box]
What has the student done wrong, and how might she get full marks?
Let . Recall the trace of a matrix is the function
So it is the sum of its diagonal entries. A commonly used inner product which on is , for any . You may assume this defines a bilinear form.
Prove that is symmetric on .
Prove that is positive definite.
Calculate the angle between the two vectors in :
Find an orthonormal basis for .
[Hint for part (ii): Use the formula from Exercise 1.25 to find an expression for the entry in the matrix .]
Let be the vector space of real-valued continuous functions on , with the inner product , as in Example 3.15. Define the following vectors in :
for each .
At the end of MATH210 it was proved that the infinite sequence is orthonormal, where refers to the constant function with value 1.
Assume we have a function in as follows:
for some scalars . Then use Theorem 3.33 to express , as an integral in terms of .
[Aside: In fact, at the end of the MATH210 notes it is proved that any continuous function can be written in the above form if we let , and no longer assume it is a finite linear combination of the orthonormal sequence. This infinite series is called the Fourier series of . These ideas will also be discussed in MATH317, where the concept of “orthonormal basis” is extended to infinite-dimensional inner product spaces.]
Learning objectives for Chapter 3:
Pass Level: You should be able to…
State the definition of an inner product space, and explain all the words you use.
Convert an orthogonal sequence into an orthonormal sequence (e.g. Exer. 3.24).
Explain the purpose of the Gram-Schmidt process.
Correctly answer, with full justification, at least 50% of the true / false questions relevant to this Chapter.
First class level: You should be able to…
Write a complete solution, without referring to any notes, to at least 80% of the exercises in this Chapter, and in particular the proof questions.
Correctly answer, with full justification, all of the true / false questions relevant to this Chapter.