V4 - Subspaces


Example 1 πŸ”—

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.

Answer:

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


Example 2 πŸ”—

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.

Answer:

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


Example 3 πŸ”—

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.

Answer:

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


Example 4 πŸ”—

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.

Answer:

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


Example 5 πŸ”—

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.

Answer:

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


Example 6 πŸ”—

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.

Answer:

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


Example 7 πŸ”—

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.

Answer:

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


Example 8 πŸ”—

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.

Answer:

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


Example 9 πŸ”—

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.

Answer:

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


Example 10 πŸ”—

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.

Answer:

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


Example 11 πŸ”—

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.

Answer:

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


Example 12 πŸ”—

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.

Answer:

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


Example 13 πŸ”—

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.

Answer:

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


Example 14 πŸ”—

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.

Answer:

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


Example 15 πŸ”—

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.

Answer:

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


Example 16 πŸ”—

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.

Answer:

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


Example 17 πŸ”—

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.

Answer:

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


Example 18 πŸ”—

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.

Answer:

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


Example 19 πŸ”—

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.

Answer:

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


Example 20 πŸ”—

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.

Answer:

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


Example 21 πŸ”—

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.

Answer:

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


Example 22 πŸ”—

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.

Answer:

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


Example 23 πŸ”—

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.

Answer:

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


Example 24 πŸ”—

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.

Answer:

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


Example 25 πŸ”—

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.

Answer:

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


Example 26 πŸ”—

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.

Answer:

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


Example 27 πŸ”—

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.

Answer:

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


Example 28 πŸ”—

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.

Answer:

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


Example 29 πŸ”—

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.

Answer:

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


Example 30 πŸ”—

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.

Answer:

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


Example 31 πŸ”—

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.

Answer:

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


Example 32 πŸ”—

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.

Answer:

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


Example 33 πŸ”—

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.

Answer:

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


Example 34 πŸ”—

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.

Answer:

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


Example 35 πŸ”—

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.

Answer:

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


Example 36 πŸ”—

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.

Answer:

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


Example 37 πŸ”—

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.

Answer:

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


Example 38 πŸ”—

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.

Answer:

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


Example 39 πŸ”—

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.

Answer:

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


Example 40 πŸ”—

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.

Answer:

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


Example 41 πŸ”—

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.

Answer:

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


Example 42 πŸ”—

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.

Answer:

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


Example 43 πŸ”—

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.

Answer:

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


Example 44 πŸ”—

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.

Answer:

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


Example 45 πŸ”—

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.

Answer:

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


Example 46 πŸ”—

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.

Answer:

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


Example 47 πŸ”—

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.

Answer:

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


Example 48 πŸ”—

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.

Answer:

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


Example 49 πŸ”—

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.

Answer:

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


Example 50 πŸ”—

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.

Answer:

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