Exercises

1. i.

   Let . Prove that for any non-zero vector $(x,y)\in {ℝ}^2$.

2. ii.

   Prove that ${ℝ}^2$ with ${⟨\cdot ,\cdot ⟩}_A$ is an inner product space.

3. iii.

   Using ${⟨\cdot ,\cdot ⟩}_A$, find the norms and angle between the vectors and .

Determine which of these bilinear forms are inner products.

1. i.

   Let $V=ℂ$, the 2-dimensional real vector space; define $⟨x,y⟩:=\mathrm{Re}(x\overline{y})$ for all $x,y\in ℂ$. Here $\mathrm{Re}(x)$ is the real part of $x$, and $\overline{x}$ is the complex conjugate of $x$.

2. ii.

   Let $V={𝒫}_2(ℝ)$. Define $⟨p(x),q(x)⟩=p(0)q(0)$.

For each of the following subspaces of ${ℝ}^n$ find an orthogonal basis.

1. i.

   $\mathrm{span}\{(1,1,0),(0,1,1)\}\subset {ℝ}^3$.

2. ii.

   $\{(x,y,z)|\mathrm{\hspace{0.33em}2}x+y=3z\}\subset {ℝ}^3$.

3. iii.

   The row space of .

4. iv.

   The orthogonal complement of .

Let $V={𝒫}_3(ℝ)$, and define the bilinear form

for $p(x),q(x)\in {𝒫}_3(ℝ)$. This defines an inner product. Apply the Gram-Schmidt process to the basis $1,x,x^2,x^3$ to produce an orthogonal basis for ${𝒫}_3(ℝ)$.

In an inner product space, prove that an orthogonal sequence of non-zero vectors is always linearly independent.

Let $W\subset {ℝ}^n$. A student is asked to prove that ${(W^{⟂})}^{⟂}=W$, and he writes the following:

[Student box]

If $x\in W$, and $y\in W^{⟂}$, then $y\cdot x=0$, by definition of $W^{⟂}$.
But we know $y\cdot x=x\cdot y$, and therefore
$x\in {(W^{⟂})}^{⟂}:=\{z\in {ℝ}^n|⟨z,y⟩=0\text{ for all }y\in W^{⟂}\}.$
Hence $W={(W^{⟂})}^{⟂}$.

[End of Student box]

What has the student done wrong, and how might he get full marks?

Prove Theorem 3.6.

[Hint: Use that the $(i,j)$ coordinate of a matrix is equal to ${\overrightarrow{e_i}}^{T}A\overrightarrow{e_j}$.]

A student is asked to prove Theorem 3.11, and she writes the following:

[Student box]

Assume that ${⟨\cdot ,\cdot ⟩}_A$ is symmetric.

Notice that for any standard basis vectors $\overrightarrow{e_i},\overrightarrow{e_j}$ we have that

So ${⟨\overrightarrow{e_i},\overrightarrow{e_j}⟩}_A={⟨\overrightarrow{e_j},\overrightarrow{e_i}⟩}_A$ implies that ${[A]}_{i,j}={[A]}_{j,i}$.

Therefore $A$ is a symmetric matrix.

[End of Student box]

What has the student done wrong, and how might she get full marks?

Let $V:={\mathrm{M}}_n(ℝ)$. 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 ${\mathrm{M}}_n(ℝ)$ is $⟨A,B⟩:=\mathrm{tr}(A{B}^T)$, for any $A,B\in {\mathrm{M}}_n(ℝ)$. You may assume this defines a bilinear form.

1. i.

   Prove that $⟨\cdot ,\cdot ⟩$ is symmetric on $V$.

2. ii.

   Prove that $⟨\cdot ,\cdot ⟩$ is positive definite.

3. iii.

   Calculate the angle between the two vectors in ${\mathrm{M}}_3(ℝ)$:

4. iv.

   Find an orthonormal basis for ${\mathrm{M}}_2(ℝ)$.

[Hint for part (ii): Use the formula from Exercise 1.25 to find an expression for the $(i,i)$ entry in the matrix $A{B}^T$.]

Let $V$ be the vector space of real-valued continuous functions on $[0,1]$, with the inner product $⟨f,g⟩={\int }_0^{1}f(t)g(t)𝑑t$, as in Example 3.15. Define the following vectors in $V$:

1. $f_n(t)=\sqrt{2}\cos (2\pi nt)$

2. $g_n(t)=\sqrt{2}\sin (2\pi nt)$

for each $n\ge 1$.

At the end of MATH210 it was proved that the infinite sequence $(1,f_1,g_1,f_2,g_2,\mathrm{\cdots })$ is orthonormal, where $1$ refers to the constant function with value 1.

Assume we have a function in $V$ as follows:

for some scalars ${\alpha }_n,{\beta }_n\in ℝ$. Then use Theorem 3.33 to express ${\alpha }_n$, ${\beta }_n$ as an integral in terms of $f$.

[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 $r=\mathrm{\infty }$, and no longer assume it is a finite linear combination of the orthonormal sequence. This infinite series is called the Fourier series of $f$. These ideas will also be discussed in MATH317, where the concept of “orthonormal basis” is extended to infinite-dimensional inner product spaces.]