V4 - Subspaces

Example 1 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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.