Inner Product Spaces

Exercise 1

In each part, determine whether the set of vectors is orthogonal and whether it is orthonormal with respect to the Euclidean inner product on ℝ².

1. $ (0, 1), (2, 0) $
2. $(- \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}), (\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}})$
3. $(- \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}), (\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}})$
4. $ (0, 0), (0, 1) $

Exercise 2

In each part, determine whether the set of vectors is orthogonal and whether it is orthonormal with respect to the Euclidean inner product on $\mathbb{R}^2$.

1. $(\frac{1}{2}, 0, \frac{1}{2}), (\frac{1}{3}, \frac{1}{3}, -\frac{1}{3}), (-\frac{1}{2}, 0, \frac{1}{2})$
2. $(\frac{2}{3}, \frac{1}{3}, \frac{1}{3}), (\frac{1}{3}, \frac{2}{3}, \frac{1}{3}), (\frac{1}{3}, \frac{1}{3}, \frac{2}{3})$
3. $ (1, 0, 0), (0, \frac{1}{2}, \frac{1}{2}), (0, 0, 1) $
4. $(\frac{1}{\sqrt{6}}, \frac{1}{\sqrt{6}}, -\frac{2}{\sqrt{6}}), (\frac{1}{\sqrt{2}}, -\frac{1}{\sqrt{2}}, 0) $

Exercise 3

In each part, determine whether the set of vectors is orthogonal with respect to the standard inner product on $P_2$.

1. $p_1(x) = \frac{1}{3}x + \frac{1}{3}x^2$, $p_2(x) = \frac{2}{3} + \frac{1}{3}x - \frac{2}{3}x^2$, $p_3(x) = \frac{1}{3} + \frac{2}{3}x + \frac{1}{3}x^2 $
2. $p_1(x) = 1$, $p_2(x) = \frac{1}{\sqrt{2}}x + \frac{1}{\sqrt{2}}x^2$, $p_3(x) = x^2$

Exercise 4

In each part, determine whether the set of vectors is orthogonal with respect to the standard inner product on $M_{22}$.

$$ \left[\begin{array}{ll}0 & 0 \\ 1 & 0\end{array}\right], \left[\begin{array}{ll}0 & 0 \\ 0 & 1\end{array}\right], \left[\begin{array}{ll}0 & 1 \\ 0 & 0\end{array}\right], \left[\begin{array}{ll}1 & 0 \\ 0 & 0\end{array}\right] $$

2.

$$ \left[\begin{array}{cc}1 & 0 \\ 0 & 0\end{array}\right], \left[\begin{array}{cc}0 & 1 \\ 1 & 0\end{array}\right], \left[\begin{array}{cc}0 & 0 \\ 0 & 1\end{array}\right], \left[\begin{array}{cc}1 & 0 \\ 0 & -1\end{array}\right] $$

Exercise 5

Show that the column vectors of $A$ form an orthogonal basis for the column space of $A$ with respect to the Euclidean inner product, and then find an orthonormal basis for that column space.

$$ A = \left[\begin{array}{ccc}1 & 2 & 0 \\ 0 & 0 & 5 \\ -1 & 2 & 0\end{array}\right] $$

Exercise 6

Show that the column vectors of $A$ form an orthogonal basis for the column space of $A$ with respect to the Euclidean inner product, and then find an orthonormal basis for that column space.

$$ A = \left[\begin{array}{ccc}1 & 1 & 2 \\ 2 & 1 & 1 \\ 3 & 2 & 3\end{array}\right] $$

Exercise 7

Verify that the vectors

$$ \mathbf{v}_1 = \left(-\frac{2}{3}, \frac{4}{3}, 0\right), \mathbf{v}_2 = \left(\frac{4}{3}, \frac{5}{3}, 0\right), \mathbf{v}_3 = (0, 0, 1) $$

form an orthonormal basis for $\mathbb{R}^2$ with respect to the Euclidean inner product. Express $\mathbf{u} = (-1, -2, 2)$ as a linear combination of $\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3$.

Exercise 8

Express $\mathbf{u} = (3, -7, 4)$ as a linear combination of the vectors from Exercise 7.

Exercise 9

Verify that the vectors

$$ \mathbf{v}_1 = (2, -2, 1), \mathbf{v}_2 = (2, 1, -2), \mathbf{v}_3 = (1, 2, 2) $$

form an orthogonal basis for $\mathbb{R}^3$ with respect to the Euclidean inner product. Express $\mathbf{u} = (-1, 0, 2)$ as a linear combination of these vectors.

Exercise 10

Verify that the vectors

$$ \mathbf{v}_1 = (-1, -1, 2, -1), \mathbf{v}_2 = (-2, 2, 3, 2), \mathbf{v}_3 = (1, 2, 0, -1), \mathbf{v}_4 = (1, 0, 0, 1) $$

form an orthogonal basis for $\mathbb{R}^4$. Express $\mathbf{u} = (1, 1, 1, 1)$ as a linear combination of these vectors.

Exercise 11

Find the coordinate vector $(u)_S$ for the vector and basis:

$$ \mathbf{u} = (4, 3, 6), \mathbf{v}_1 = (1, -2, 2), \mathbf{v}_2 = (2, 1, 1), \mathbf{v}_3 = (0, 3, 3) $$

Exercise 12

Find the coordinate vector $(u)_S$ for:

$$ \mathbf{u} = (1, -1, 3), \mathbf{v}_1 = (2, 0, 1), \mathbf{v}_2 = (0, 1, 0), \mathbf{v}_3 = (-1, 0, 2) $$

Exercise 13

Find the coordinate vector $(u)_S$ for:

$$ \mathbf{u} = (-1, 1, 2, 1), \mathbf{v}_1 = (1, 1, 1, 1), \mathbf{v}_2 = (1, 1, -1, -1), \mathbf{v}_3 = (1, -1, 1, -1), \mathbf{v}_4 = (1, -1, -1, 1) $$

Exercise 14

Find the coordinate vector $(u)_S$ for:

$$ \mathbf{u} = (1, 1, 1, 1), \mathbf{v}_1 = (-1, -1, 2, -1), \mathbf{v}_2 = (-2, 2, 3, 2), \mathbf{v}_3 = (1, 2, 0, -1), \mathbf{v}_4 = (1, 0, 0, 1) $$

Exercise 15

Let $\mathbb{R}^2$ have the Euclidean inner product.

1. Find the orthogonal projection of $\mathbf{u}$ onto the line spanned by $\mathbf{v}$.
2. Find the component of $\mathbf{u}$ orthogonal to that line and confirm orthogonality.

$\mathbf{u} = (-1, 6); \mathbf{v} = (3, 5)$

Exercise 16

Let $\mathbb{R}^2$ have the Euclidean inner product.

1. Find the orthogonal projection of $\mathbf{u}$ onto the line spanned by $\mathbf{v}$.
2. Find the component of $\mathbf{u}$ orthogonal to that line and confirm orthogonality.

$\mathbf{u} = (2, 3); \mathbf{v} = (\frac{5}{13}, \frac{12}{13})$

Exercise 17

Let $\mathbb{R}^2$ have the Euclidean inner product.

1. Find the orthogonal projection of $\mathbf{u}$ onto the line spanned by $\mathbf{v}$.
2. Find the component of $\mathbf{u}$ orthogonal to that line and confirm orthogonality.

$\mathbf{u} = (2, 3); \mathbf{v} = (1, 2)$

Exercise 18

Let $\mathbb{R}^2$ have the Euclidean inner product.

1. Find the orthogonal projection of $\mathbf{u}$ onto the line spanned by $\mathbf{v}$.
2. Find the component of $\mathbf{u}$ orthogonal to that line and confirm orthogonality.

$\mathbf{u} = (3, -1); \mathbf{v} = (3, 4)$

Exercise 19

Let $\mathbb{R}^3$ have the Euclidean inner product.

1. Find the orthogonal projection of $\mathbf{u}$ onto the plane spanned by $\mathbf{v}_1, \mathbf{v}_2$.
2. Find the orthogonal component and confirm orthogonality.

$$ \mathbf{u} = (4, 2, 1) $$

$$ \mathbf{v}_1 = (\frac{1}{3}, \frac{2}{3}, -\frac{1}{3}) $$

$$ \mathbf{v}_2 = (\frac{2}{3}, \frac{1}{3}, \frac{2}{3}) $$

Exercise 20

Let $\mathbb{R}^3$ have the Euclidean inner product.

1. Find the orthogonal projection of $\mathbf{u}$ onto the plane spanned by $ \mathbf{v}_1, \mathbf{v}_2 $.
2. Find the orthogonal component and confirm orthogonality.

$$ \mathbf{u} = (3, -1, 2) $$

$$ \mathbf{v}_1 = (-\frac{1}{\sqrt{6}}, -\frac{2}{\sqrt{6}}, \frac{1}{\sqrt{6}}), $$