## V4 - Subspaces

#### Example 1 🔗

Consider the following two sets of Euclidean vectors:

\begin{align*} \left\{ \left[\begin{array}{c} x \\ y \\ z \end{array}\right] \middle|\,y^{2} - 7 \, z^{2} = 4 \, x\right\} & & \left\{ \left[\begin{array}{c} x \\ y \\ z \end{array}\right] \middle|\,3 \, y = 7 \, x - 2 \, z\right\} \\ \end{align*}

Explain why one of these sets is a subspace of $$\mathbb{R}^ 3$$ and one is not.

$$U$$ is a subspace of $$\mathbb{R}^ 3$$ and $$W$$ is not.

#### Example 2 🔗

Consider the following two sets of Euclidean vectors:

\begin{align*} \left\{ \left[\begin{array}{c} x \\ y \end{array}\right] \middle|\,x^{2} = 4 \, y\right\} & & \left\{ \left[\begin{array}{c} x \\ y \end{array}\right] \middle|\,3 \, x + 3 \, y = 0\right\} \\ \end{align*}

Explain why one of these sets is a subspace of $$\mathbb{R}^ 2$$ and one is not.

$$U$$ is a subspace of $$\mathbb{R}^ 2$$ and $$W$$ is not.

#### Example 3 🔗

Consider the following two sets of Euclidean vectors:

\begin{align*} \left\{ \left[\begin{array}{c} x \\ y \end{array}\right] \middle|\,2 \, x + 4 \, y = 0\right\} & & \left\{ \left[\begin{array}{c} x \\ y \end{array}\right] \middle|\,x^{3} + 5 \, y = 0\right\} \\ \end{align*}

Explain why one of these sets is a subspace of $$\mathbb{R}^ 2$$ and one is not.

$$U$$ is a subspace of $$\mathbb{R}^ 2$$ and $$W$$ is not.

#### Example 4 🔗

Consider the following two sets of Euclidean vectors:

\begin{align*} \left\{ \left[\begin{array}{c} x \\ y \\ z \\ w \end{array}\right] \middle|\,w + 7 \, x = 4 \, y + 5 \, z\right\} & & \left\{ \left[\begin{array}{c} x \\ y \\ z \\ w \end{array}\right] \middle|\,3 \, x = z^{3} - 4 \, w + 2 \, y\right\} \\ \end{align*}

Explain why one of these sets is a subspace of $$\mathbb{R}^ 4$$ and one is not.

$$U$$ is a subspace of $$\mathbb{R}^ 4$$ and $$W$$ is not.

#### Example 5 🔗

Consider the following two sets of Euclidean vectors:

\begin{align*} \left\{ \left[\begin{array}{c} x \\ y \\ z \end{array}\right] \middle|\,4 \, x + 5 \, y = 5 \, z\right\} & & \left\{ \left[\begin{array}{c} x \\ y \\ z \end{array}\right] \middle|\,y^{2} - 6 \, z^{2} = 6 \, x\right\} \\ \end{align*}

Explain why one of these sets is a subspace of $$\mathbb{R}^ 3$$ and one is not.

$$U$$ is a subspace of $$\mathbb{R}^ 3$$ and $$W$$ is not.

#### Example 6 🔗

Consider the following two sets of Euclidean vectors:

\begin{align*} \left\{ \left[\begin{array}{c} x \\ y \\ z \end{array}\right] \middle|\,x + 7 \, y + 5 \, z = 0\right\} & & \left\{ \left[\begin{array}{c} x \\ y \\ z \end{array}\right] \middle|\,y^{2} - 7 \, z^{2} = 4 \, x\right\} \\ \end{align*}

Explain why one of these sets is a subspace of $$\mathbb{R}^ 3$$ and one is not.

$$U$$ is a subspace of $$\mathbb{R}^ 3$$ and $$W$$ is not.

#### Example 7 🔗

Consider the following two sets of Euclidean vectors:

\begin{align*} \left\{ \left[\begin{array}{c} x \\ y \\ z \end{array}\right] \middle|\,3 \, x + 2 \, y = 2 \, z\right\} & & \left\{ \left[\begin{array}{c} x \\ y \\ z \end{array}\right] \middle|\,x^{3} + 2 \, y + 2 \, z = 0\right\} \\ \end{align*}

Explain why one of these sets is a subspace of $$\mathbb{R}^ 3$$ and one is not.

$$U$$ is a subspace of $$\mathbb{R}^ 3$$ and $$W$$ is not.

#### Example 8 🔗

Consider the following two sets of Euclidean vectors:

\begin{align*} \left\{ \left[\begin{array}{c} x \\ y \end{array}\right] \middle|\,x^{2} = y\right\} & & \left\{ \left[\begin{array}{c} x \\ y \end{array}\right] \middle|\,2 \, x = 3 \, y\right\} \\ \end{align*}

Explain why one of these sets is a subspace of $$\mathbb{R}^ 2$$ and one is not.

$$U$$ is a subspace of $$\mathbb{R}^ 2$$ and $$W$$ is not.

#### Example 9 🔗

Consider the following two sets of Euclidean vectors:

\begin{align*} \left\{ \left[\begin{array}{c} x \\ y \\ z \end{array}\right] \middle|\,2 \, x + 5 \, y = 5 \, z\right\} & & \left\{ \left[\begin{array}{c} x \\ y \\ z \end{array}\right] \middle|\,2 \, x^{2} y + 5 \, z = 0\right\} \\ \end{align*}

Explain why one of these sets is a subspace of $$\mathbb{R}^ 3$$ and one is not.

$$U$$ is a subspace of $$\mathbb{R}^ 3$$ and $$W$$ is not.

#### Example 10 🔗

Consider the following two sets of Euclidean vectors:

\begin{align*} \left\{ \left[\begin{array}{c} x \\ y \\ z \end{array}\right] \middle|\,7 \, x = z^{3} + 2 \, y\right\} & & \left\{ \left[\begin{array}{c} x \\ y \\ z \end{array}\right] \middle|\,y = x - 2 \, z\right\} \\ \end{align*}

Explain why one of these sets is a subspace of $$\mathbb{R}^ 3$$ and one is not.

$$U$$ is a subspace of $$\mathbb{R}^ 3$$ and $$W$$ is not.

#### Example 11 🔗

Consider the following two sets of Euclidean vectors:

\begin{align*} \left\{ \left[\begin{array}{c} x \\ y \\ z \end{array}\right] \middle|\,6 \, x = z^{2} + 3 \, y\right\} & & \left\{ \left[\begin{array}{c} x \\ y \\ z \end{array}\right] \middle|\,3 \, x + 7 \, y = 3 \, z\right\} \\ \end{align*}

Explain why one of these sets is a subspace of $$\mathbb{R}^ 3$$ and one is not.

$$U$$ is a subspace of $$\mathbb{R}^ 3$$ and $$W$$ is not.

#### Example 12 🔗

Consider the following two sets of Euclidean vectors:

\begin{align*} \left\{ \left[\begin{array}{c} x \\ y \\ z \end{array}\right] \middle|\,x^{2} + 3 \, y + 5 \, z = 0\right\} & & \left\{ \left[\begin{array}{c} x \\ y \\ z \end{array}\right] \middle|\,3 \, x = 2 \, y + 4 \, z\right\} \\ \end{align*}

Explain why one of these sets is a subspace of $$\mathbb{R}^ 3$$ and one is not.

$$U$$ is a subspace of $$\mathbb{R}^ 3$$ and $$W$$ is not.

#### Example 13 🔗

Consider the following two sets of Euclidean vectors:

\begin{align*} \left\{ \left[\begin{array}{c} x \\ y \end{array}\right] \middle|\,x^{2} + 7 \, y = 0\right\} & & \left\{ \left[\begin{array}{c} x \\ y \end{array}\right] \middle|\,2 \, x + 2 \, y = 0\right\} \\ \end{align*}

Explain why one of these sets is a subspace of $$\mathbb{R}^ 2$$ and one is not.

$$U$$ is a subspace of $$\mathbb{R}^ 2$$ and $$W$$ is not.

#### Example 14 🔗

Consider the following two sets of Euclidean vectors:

\begin{align*} \left\{ \left[\begin{array}{c} x \\ y \\ z \end{array}\right] \middle|\,3 \, y = 2 \, x - 4 \, z\right\} & & \left\{ \left[\begin{array}{c} x \\ y \\ z \end{array}\right] \middle|\,y^{2} + 3 \, x = 3 \, z\right\} \\ \end{align*}

Explain why one of these sets is a subspace of $$\mathbb{R}^ 3$$ and one is not.

$$U$$ is a subspace of $$\mathbb{R}^ 3$$ and $$W$$ is not.

#### Example 15 🔗

Consider the following two sets of Euclidean vectors:

\begin{align*} \left\{ \left[\begin{array}{c} x \\ y \\ z \end{array}\right] \middle|\,x + 7 \, y = 3 \, z\right\} & & \left\{ \left[\begin{array}{c} x \\ y \\ z \end{array}\right] \middle|\,5 \, x^{2} y + 4 \, z = 0\right\} \\ \end{align*}

Explain why one of these sets is a subspace of $$\mathbb{R}^ 3$$ and one is not.

$$U$$ is a subspace of $$\mathbb{R}^ 3$$ and $$W$$ is not.

#### Example 16 🔗

Consider the following two sets of Euclidean vectors:

\begin{align*} \left\{ \left[\begin{array}{c} x \\ y \\ z \\ w \end{array}\right] \middle|\,7 \, w + 6 \, x = 3 \, z\right\} & & \left\{ \left[\begin{array}{c} x \\ y \\ z \\ w \end{array}\right] \middle|\,x^{3} + 4 \, y + 2 \, z = 5 \, w\right\} \\ \end{align*}

Explain why one of these sets is a subspace of $$\mathbb{R}^ 4$$ and one is not.

$$U$$ is a subspace of $$\mathbb{R}^ 4$$ and $$W$$ is not.

#### Example 17 🔗

Consider the following two sets of Euclidean vectors:

\begin{align*} \left\{ \left[\begin{array}{c} x \\ y \end{array}\right] \middle|\,3 \, x = 3 \, y\right\} & & \left\{ \left[\begin{array}{c} x \\ y \end{array}\right] \middle|\,x^{2} = y\right\} \\ \end{align*}

Explain why one of these sets is a subspace of $$\mathbb{R}^ 2$$ and one is not.

$$U$$ is a subspace of $$\mathbb{R}^ 2$$ and $$W$$ is not.

#### Example 18 🔗

Consider the following two sets of Euclidean vectors:

\begin{align*} \left\{ \left[\begin{array}{c} x \\ y \\ z \\ w \end{array}\right] \middle|\,6 \, x + 6 \, y + 4 \, z = 2 \, w\right\} & & \left\{ \left[\begin{array}{c} x \\ y \\ z \\ w \end{array}\right] \middle|\,x^{2} + 7 \, y + 3 \, z = 3 \, w\right\} \\ \end{align*}

Explain why one of these sets is a subspace of $$\mathbb{R}^ 4$$ and one is not.

$$U$$ is a subspace of $$\mathbb{R}^ 4$$ and $$W$$ is not.

#### Example 19 🔗

Consider the following two sets of Euclidean vectors:

\begin{align*} \left\{ \left[\begin{array}{c} x \\ y \\ z \\ w \end{array}\right] \middle|\,3 \, x + 3 \, y + 2 \, z = 2 \, w\right\} & & \left\{ \left[\begin{array}{c} x \\ y \\ z \\ w \end{array}\right] \middle|\,x^{2} + 5 \, y + 3 \, z = 2 \, w\right\} \\ \end{align*}

Explain why one of these sets is a subspace of $$\mathbb{R}^ 4$$ and one is not.

$$U$$ is a subspace of $$\mathbb{R}^ 4$$ and $$W$$ is not.

#### Example 20 🔗

Consider the following two sets of Euclidean vectors:

\begin{align*} \left\{ \left[\begin{array}{c} x \\ y \\ z \end{array}\right] \middle|\,x^{3} + 7 \, y + 5 \, z = 0\right\} & & \left\{ \left[\begin{array}{c} x \\ y \\ z \end{array}\right] \middle|\,7 \, x + 6 \, y = 5 \, z\right\} \\ \end{align*}

Explain why one of these sets is a subspace of $$\mathbb{R}^ 3$$ and one is not.

$$U$$ is a subspace of $$\mathbb{R}^ 3$$ and $$W$$ is not.

#### Example 21 🔗

Consider the following two sets of Euclidean vectors:

\begin{align*} \left\{ \left[\begin{array}{c} x \\ y \end{array}\right] \middle|\,5 \, x = 3 \, y\right\} & & \left\{ \left[\begin{array}{c} x \\ y \end{array}\right] \middle|\,x^{2} = 3 \, y\right\} \\ \end{align*}

Explain why one of these sets is a subspace of $$\mathbb{R}^ 2$$ and one is not.

$$U$$ is a subspace of $$\mathbb{R}^ 2$$ and $$W$$ is not.

#### Example 22 🔗

Consider the following two sets of Euclidean vectors:

\begin{align*} \left\{ \left[\begin{array}{c} x \\ y \end{array}\right] \middle|\,x^{3} = y\right\} & & \left\{ \left[\begin{array}{c} x \\ y \end{array}\right] \middle|\,6 \, x + y = 0\right\} \\ \end{align*}

Explain why one of these sets is a subspace of $$\mathbb{R}^ 2$$ and one is not.

$$U$$ is a subspace of $$\mathbb{R}^ 2$$ and $$W$$ is not.

#### Example 23 🔗

Consider the following two sets of Euclidean vectors:

\begin{align*} \left\{ \left[\begin{array}{c} x \\ y \\ z \end{array}\right] \middle|\,y^{3} + 4 \, x = 4 \, z\right\} & & \left\{ \left[\begin{array}{c} x \\ y \\ z \end{array}\right] \middle|\,6 \, x = 3 \, y + 5 \, z\right\} \\ \end{align*}

Explain why one of these sets is a subspace of $$\mathbb{R}^ 3$$ and one is not.

$$U$$ is a subspace of $$\mathbb{R}^ 3$$ and $$W$$ is not.

#### Example 24 🔗

Consider the following two sets of Euclidean vectors:

\begin{align*} \left\{ \left[\begin{array}{c} x \\ y \\ z \\ w \end{array}\right] \middle|\,4 \, x^{2} y + 3 \, w z = 0\right\} & & \left\{ \left[\begin{array}{c} x \\ y \\ z \\ w \end{array}\right] \middle|\,3 \, w + 2 \, x = 5 \, y + 4 \, z\right\} \\ \end{align*}

Explain why one of these sets is a subspace of $$\mathbb{R}^ 4$$ and one is not.

$$U$$ is a subspace of $$\mathbb{R}^ 4$$ and $$W$$ is not.

#### Example 25 🔗

Consider the following two sets of Euclidean vectors:

\begin{align*} \left\{ \left[\begin{array}{c} x \\ y \\ z \\ w \end{array}\right] \middle|\,4 \, w + 3 \, y = 4 \, x - 3 \, z\right\} & & \left\{ \left[\begin{array}{c} x \\ y \\ z \\ w \end{array}\right] \middle|\,6 \, x^{2} y + 4 \, w z = 0\right\} \\ \end{align*}

Explain why one of these sets is a subspace of $$\mathbb{R}^ 4$$ and one is not.

$$U$$ is a subspace of $$\mathbb{R}^ 4$$ and $$W$$ is not.

#### Example 26 🔗

Consider the following two sets of Euclidean vectors:

\begin{align*} \left\{ \left[\begin{array}{c} x \\ y \\ z \end{array}\right] \middle|\,y^{2} - 2 \, z^{2} = 6 \, x\right\} & & \left\{ \left[\begin{array}{c} x \\ y \\ z \end{array}\right] \middle|\,7 \, x = 7 \, y + 3 \, z\right\} \\ \end{align*}

Explain why one of these sets is a subspace of $$\mathbb{R}^ 3$$ and one is not.

$$U$$ is a subspace of $$\mathbb{R}^ 3$$ and $$W$$ is not.

#### Example 27 🔗

Consider the following two sets of Euclidean vectors:

\begin{align*} \left\{ \left[\begin{array}{c} x \\ y \end{array}\right] \middle|\,2 \, x = 2 \, y\right\} & & \left\{ \left[\begin{array}{c} x \\ y \end{array}\right] \middle|\,x^{2} + 3 \, y = 0\right\} \\ \end{align*}

Explain why one of these sets is a subspace of $$\mathbb{R}^ 2$$ and one is not.

$$U$$ is a subspace of $$\mathbb{R}^ 2$$ and $$W$$ is not.

#### Example 28 🔗

Consider the following two sets of Euclidean vectors:

\begin{align*} \left\{ \left[\begin{array}{c} x \\ y \end{array}\right] \middle|\,x^{3} = 5 \, y\right\} & & \left\{ \left[\begin{array}{c} x \\ y \end{array}\right] \middle|\,7 \, x = 4 \, y\right\} \\ \end{align*}

Explain why one of these sets is a subspace of $$\mathbb{R}^ 2$$ and one is not.

$$U$$ is a subspace of $$\mathbb{R}^ 2$$ and $$W$$ is not.

#### Example 29 🔗

Consider the following two sets of Euclidean vectors:

\begin{align*} \left\{ \left[\begin{array}{c} x \\ y \\ z \\ w \end{array}\right] \middle|\,6 \, w + 4 \, x = 4 \, z\right\} & & \left\{ \left[\begin{array}{c} x \\ y \\ z \\ w \end{array}\right] \middle|\,x^{3} + 4 \, y + 5 \, z = 5 \, w\right\} \\ \end{align*}

Explain why one of these sets is a subspace of $$\mathbb{R}^ 4$$ and one is not.

$$U$$ is a subspace of $$\mathbb{R}^ 4$$ and $$W$$ is not.

#### Example 30 🔗

Consider the following two sets of Euclidean vectors:

\begin{align*} \left\{ \left[\begin{array}{c} x \\ y \\ z \end{array}\right] \middle|\,5 \, x + 3 \, y = 2 \, z\right\} & & \left\{ \left[\begin{array}{c} x \\ y \\ z \end{array}\right] \middle|\,7 \, x = z^{3} + 2 \, y\right\} \\ \end{align*}

Explain why one of these sets is a subspace of $$\mathbb{R}^ 3$$ and one is not.

$$U$$ is a subspace of $$\mathbb{R}^ 3$$ and $$W$$ is not.

#### Example 31 🔗

Consider the following two sets of Euclidean vectors:

\begin{align*} \left\{ \left[\begin{array}{c} x \\ y \end{array}\right] \middle|\,x^{3} = 6 \, y\right\} & & \left\{ \left[\begin{array}{c} x \\ y \end{array}\right] \middle|\,3 \, x = 5 \, y\right\} \\ \end{align*}

Explain why one of these sets is a subspace of $$\mathbb{R}^ 2$$ and one is not.

$$U$$ is a subspace of $$\mathbb{R}^ 2$$ and $$W$$ is not.

#### Example 32 🔗

Consider the following two sets of Euclidean vectors:

\begin{align*} \left\{ \left[\begin{array}{c} x \\ y \end{array}\right] \middle|\,2 \, x y^{3} = 0\right\} & & \left\{ \left[\begin{array}{c} x \\ y \end{array}\right] \middle|\,4 \, x + 7 \, y = 0\right\} \\ \end{align*}

Explain why one of these sets is a subspace of $$\mathbb{R}^ 2$$ and one is not.

$$U$$ is a subspace of $$\mathbb{R}^ 2$$ and $$W$$ is not.

#### Example 33 🔗

Consider the following two sets of Euclidean vectors:

\begin{align*} \left\{ \left[\begin{array}{c} x \\ y \\ z \end{array}\right] \middle|\,5 \, x = 3 \, y + 3 \, z\right\} & & \left\{ \left[\begin{array}{c} x \\ y \\ z \end{array}\right] \middle|\,x^{3} + 2 \, y + 2 \, z = 0\right\} \\ \end{align*}

Explain why one of these sets is a subspace of $$\mathbb{R}^ 3$$ and one is not.

$$U$$ is a subspace of $$\mathbb{R}^ 3$$ and $$W$$ is not.

#### Example 34 🔗

Consider the following two sets of Euclidean vectors:

\begin{align*} \left\{ \left[\begin{array}{c} x \\ y \\ z \\ w \end{array}\right] \middle|\,4 \, x + 4 \, y + 2 \, z = 3 \, w\right\} & & \left\{ \left[\begin{array}{c} x \\ y \\ z \\ w \end{array}\right] \middle|\,2 \, x^{3} y + 4 \, w z = 0\right\} \\ \end{align*}

Explain why one of these sets is a subspace of $$\mathbb{R}^ 4$$ and one is not.

$$U$$ is a subspace of $$\mathbb{R}^ 4$$ and $$W$$ is not.

#### Example 35 🔗

Consider the following two sets of Euclidean vectors:

\begin{align*} \left\{ \left[\begin{array}{c} x \\ y \\ z \end{array}\right] \middle|\,7 \, x + 3 \, y = 3 \, z\right\} & & \left\{ \left[\begin{array}{c} x \\ y \\ z \end{array}\right] \middle|\,x^{3} + 3 \, y + 4 \, z = 0\right\} \\ \end{align*}

Explain why one of these sets is a subspace of $$\mathbb{R}^ 3$$ and one is not.

$$U$$ is a subspace of $$\mathbb{R}^ 3$$ and $$W$$ is not.

#### Example 36 🔗

Consider the following two sets of Euclidean vectors:

\begin{align*} \left\{ \left[\begin{array}{c} x \\ y \\ z \end{array}\right] \middle|\,y = x - 5 \, z\right\} & & \left\{ \left[\begin{array}{c} x \\ y \\ z \end{array}\right] \middle|\,x^{3} + 6 \, y + 5 \, z = 0\right\} \\ \end{align*}

Explain why one of these sets is a subspace of $$\mathbb{R}^ 3$$ and one is not.

$$U$$ is a subspace of $$\mathbb{R}^ 3$$ and $$W$$ is not.

#### Example 37 🔗

Consider the following two sets of Euclidean vectors:

\begin{align*} \left\{ \left[\begin{array}{c} x \\ y \\ z \\ w \end{array}\right] \middle|\,2 \, w + 2 \, x = y + 2 \, z\right\} & & \left\{ \left[\begin{array}{c} x \\ y \\ z \\ w \end{array}\right] \middle|\,y^{2} + 7 \, x = 3 \, z\right\} \\ \end{align*}

Explain why one of these sets is a subspace of $$\mathbb{R}^ 4$$ and one is not.

$$U$$ is a subspace of $$\mathbb{R}^ 4$$ and $$W$$ is not.

#### Example 38 🔗

Consider the following two sets of Euclidean vectors:

\begin{align*} \left\{ \left[\begin{array}{c} x \\ y \end{array}\right] \middle|\,7 \, x = 5 \, y\right\} & & \left\{ \left[\begin{array}{c} x \\ y \end{array}\right] \middle|\,x^{2} + 5 \, y = 0\right\} \\ \end{align*}

Explain why one of these sets is a subspace of $$\mathbb{R}^ 2$$ and one is not.

$$U$$ is a subspace of $$\mathbb{R}^ 2$$ and $$W$$ is not.

#### Example 39 🔗

Consider the following two sets of Euclidean vectors:

\begin{align*} \left\{ \left[\begin{array}{c} x \\ y \end{array}\right] \middle|\,2 \, x + 5 \, y = 0\right\} & & \left\{ \left[\begin{array}{c} x \\ y \end{array}\right] \middle|\,2 \, x y^{2} = 0\right\} \\ \end{align*}

Explain why one of these sets is a subspace of $$\mathbb{R}^ 2$$ and one is not.

$$U$$ is a subspace of $$\mathbb{R}^ 2$$ and $$W$$ is not.

#### Example 40 🔗

Consider the following two sets of Euclidean vectors:

\begin{align*} \left\{ \left[\begin{array}{c} x \\ y \\ z \end{array}\right] \middle|\,2 \, x = 2 \, y + 4 \, z\right\} & & \left\{ \left[\begin{array}{c} x \\ y \\ z \end{array}\right] \middle|\,6 \, x = z^{3} + 2 \, y\right\} \\ \end{align*}

Explain why one of these sets is a subspace of $$\mathbb{R}^ 3$$ and one is not.

$$U$$ is a subspace of $$\mathbb{R}^ 3$$ and $$W$$ is not.

#### Example 41 🔗

Consider the following two sets of Euclidean vectors:

\begin{align*} \left\{ \left[\begin{array}{c} x \\ y \\ z \\ w \end{array}\right] \middle|\,x^{3} + 6 \, y + 4 \, z = 3 \, w\right\} & & \left\{ \left[\begin{array}{c} x \\ y \\ z \\ w \end{array}\right] \middle|\,5 \, w + 4 \, x = 2 \, y + 2 \, z\right\} \\ \end{align*}

Explain why one of these sets is a subspace of $$\mathbb{R}^ 4$$ and one is not.

$$U$$ is a subspace of $$\mathbb{R}^ 4$$ and $$W$$ is not.

#### Example 42 🔗

Consider the following two sets of Euclidean vectors:

\begin{align*} \left\{ \left[\begin{array}{c} x \\ y \\ z \end{array}\right] \middle|\,5 \, y = 5 \, x - 2 \, z\right\} & & \left\{ \left[\begin{array}{c} x \\ y \\ z \end{array}\right] \middle|\,x^{2} y + 5 \, z = 0\right\} \\ \end{align*}

Explain why one of these sets is a subspace of $$\mathbb{R}^ 3$$ and one is not.

$$U$$ is a subspace of $$\mathbb{R}^ 3$$ and $$W$$ is not.

#### Example 43 🔗

Consider the following two sets of Euclidean vectors:

\begin{align*} \left\{ \left[\begin{array}{c} x \\ y \end{array}\right] \middle|\,4 \, x = 3 \, y\right\} & & \left\{ \left[\begin{array}{c} x \\ y \end{array}\right] \middle|\,7 \, x y^{2} = 0\right\} \\ \end{align*}

Explain why one of these sets is a subspace of $$\mathbb{R}^ 2$$ and one is not.

$$U$$ is a subspace of $$\mathbb{R}^ 2$$ and $$W$$ is not.

#### Example 44 🔗

Consider the following two sets of Euclidean vectors:

\begin{align*} \left\{ \left[\begin{array}{c} x \\ y \end{array}\right] \middle|\,3 \, x = 3 \, y\right\} & & \left\{ \left[\begin{array}{c} x \\ y \end{array}\right] \middle|\,x^{2} + 7 \, y = 0\right\} \\ \end{align*}

Explain why one of these sets is a subspace of $$\mathbb{R}^ 2$$ and one is not.

$$U$$ is a subspace of $$\mathbb{R}^ 2$$ and $$W$$ is not.

#### Example 45 🔗

Consider the following two sets of Euclidean vectors:

\begin{align*} \left\{ \left[\begin{array}{c} x \\ y \\ z \end{array}\right] \middle|\,2 \, x + 6 \, y = 4 \, z\right\} & & \left\{ \left[\begin{array}{c} x \\ y \\ z \end{array}\right] \middle|\,x^{2} + 6 \, y + 3 \, z = 0\right\} \\ \end{align*}

Explain why one of these sets is a subspace of $$\mathbb{R}^ 3$$ and one is not.

$$U$$ is a subspace of $$\mathbb{R}^ 3$$ and $$W$$ is not.

#### Example 46 🔗

Consider the following two sets of Euclidean vectors:

\begin{align*} \left\{ \left[\begin{array}{c} x \\ y \\ z \end{array}\right] \middle|\,x^{2} + 7 \, y + 5 \, z = 0\right\} & & \left\{ \left[\begin{array}{c} x \\ y \\ z \end{array}\right] \middle|\,x + y + 3 \, z = 0\right\} \\ \end{align*}

Explain why one of these sets is a subspace of $$\mathbb{R}^ 3$$ and one is not.

$$U$$ is a subspace of $$\mathbb{R}^ 3$$ and $$W$$ is not.

#### Example 47 🔗

Consider the following two sets of Euclidean vectors:

\begin{align*} \left\{ \left[\begin{array}{c} x \\ y \\ z \\ w \end{array}\right] \middle|\,4 \, x + 4 \, y + 5 \, z = 2 \, w\right\} & & \left\{ \left[\begin{array}{c} x \\ y \\ z \\ w \end{array}\right] \middle|\,4 \, x = z^{3} - 5 \, w + 4 \, y\right\} \\ \end{align*}

Explain why one of these sets is a subspace of $$\mathbb{R}^ 4$$ and one is not.

$$U$$ is a subspace of $$\mathbb{R}^ 4$$ and $$W$$ is not.

#### Example 48 🔗

Consider the following two sets of Euclidean vectors:

\begin{align*} \left\{ \left[\begin{array}{c} x \\ y \\ z \end{array}\right] \middle|\,2 \, x = 7 \, y + 3 \, z\right\} & & \left\{ \left[\begin{array}{c} x \\ y \\ z \end{array}\right] \middle|\,x^{3} + 3 \, y + 3 \, z = 0\right\} \\ \end{align*}

Explain why one of these sets is a subspace of $$\mathbb{R}^ 3$$ and one is not.

$$U$$ is a subspace of $$\mathbb{R}^ 3$$ and $$W$$ is not.

#### Example 49 🔗

Consider the following two sets of Euclidean vectors:

\begin{align*} \left\{ \left[\begin{array}{c} x \\ y \\ z \\ w \end{array}\right] \middle|\,x = z^{2} - 2 \, w + 2 \, y\right\} & & \left\{ \left[\begin{array}{c} x \\ y \\ z \\ w \end{array}\right] \middle|\,4 \, w + 5 \, x = 3 \, z\right\} \\ \end{align*}

Explain why one of these sets is a subspace of $$\mathbb{R}^ 4$$ and one is not.

$$U$$ is a subspace of $$\mathbb{R}^ 4$$ and $$W$$ is not.

#### Example 50 🔗

Consider the following two sets of Euclidean vectors:

\begin{align*} \left\{ \left[\begin{array}{c} x \\ y \\ z \end{array}\right] \middle|\,6 \, x = 2 \, y + 3 \, z\right\} & & \left\{ \left[\begin{array}{c} x \\ y \\ z \end{array}\right] \middle|\,x^{2} + 5 \, y + 3 \, z = 0\right\} \\ \end{align*}

Explain why one of these sets is a subspace of $$\mathbb{R}^ 3$$ and one is not.

$$U$$ is a subspace of $$\mathbb{R}^ 3$$ and $$W$$ is not.