V4 - Subspaces


Example 1

V4 - Subspaces (ver. 1)

Consider the following two sets of Euclidean vectors:

\[U=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} 3 \, y z - x = 0 \right\} \hspace{2em} W=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} 3 \, y - z = 4 \, x \right\}.\]

Explain why one of these sets is a subspace of \(\mathbb{R}^3\), and why the other is not.

Answer.

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


Example 2

V4 - Subspaces (ver. 2)

Consider the following two sets of Euclidean vectors:

\[U=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} 3 \, x - 2 \, y - 4 \, z = 0 \right\} \hspace{2em} W=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} 3 \, x y - 2 \, z = 0 \right\}.\]

Explain why one of these sets is a subspace of \(\mathbb{R}^3\), and why the other is not.

Answer.

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


Example 3

V4 - Subspaces (ver. 3)

Consider the following two sets of Euclidean vectors:

\[U=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} 4 \, x - 5 \, y + 3 \, z = 0 \right\} \hspace{2em} W=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} -5 \, {\left| x \right|} = 4 \, y + 3 \, z \right\}.\]

Explain why one of these sets is a subspace of \(\mathbb{R}^3\), and why the other is not.

Answer.

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


Example 4

V4 - Subspaces (ver. 4)

Consider the following two sets of Euclidean vectors:

\[U=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} 2 \, y = x - 3 \, z \right\} \hspace{2em} W=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} 2 \, x y = -3 \, z \right\}.\]

Explain why one of these sets is a subspace of \(\mathbb{R}^3\), and why the other is not.

Answer.

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


Example 5

V4 - Subspaces (ver. 5)

Consider the following two sets of Euclidean vectors:

\[U=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} -3 \, x y = -2 \, z \right\} \hspace{2em} W=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} 2 \, x - 2 \, y - 3 \, z = 0 \right\}.\]

Explain why one of these sets is a subspace of \(\mathbb{R}^3\), and why the other is not.

Answer.

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


Example 6

V4 - Subspaces (ver. 6)

Consider the following two sets of Euclidean vectors:

\[U=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} y^{3} + 5 \, x = -2 \, z \right\} \hspace{2em} W=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} 5 \, y = -2 \, x + 3 \, z \right\}.\]

Explain why one of these sets is a subspace of \(\mathbb{R}^3\), and why the other is not.

Answer.

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


Example 7

V4 - Subspaces (ver. 7)

Consider the following two sets of Euclidean vectors:

\[U=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} x^{3} + 5 \, y + 4 \, z = 0 \right\} \hspace{2em} W=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} 4 \, x + 5 \, z = 3 \, y \right\}.\]

Explain why one of these sets is a subspace of \(\mathbb{R}^3\), and why the other is not.

Answer.

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


Example 8

V4 - Subspaces (ver. 8)

Consider the following two sets of Euclidean vectors:

\[U=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} 5 \, {\left| y \right|} = 5 \, x + z \right\} \hspace{2em} W=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} 5 \, x + 5 \, y + z = 0 \right\}.\]

Explain why one of these sets is a subspace of \(\mathbb{R}^3\), and why the other is not.

Answer.

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


Example 9

V4 - Subspaces (ver. 9)

Consider the following two sets of Euclidean vectors:

\[U=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} 2 \, x = 2 \, y - 3 \, z \right\} \hspace{2em} W=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} 2 \, x z = 2 \, y \right\}.\]

Explain why one of these sets is a subspace of \(\mathbb{R}^3\), and why the other is not.

Answer.

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


Example 10

V4 - Subspaces (ver. 10)

Consider the following two sets of Euclidean vectors:

\[U=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} 3 \, x = y - z \right\} \hspace{2em} W=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} x^{2} + 3 \, z = -y \right\}.\]

Explain why one of these sets is a subspace of \(\mathbb{R}^3\), and why the other is not.

Answer.

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


Example 11

V4 - Subspaces (ver. 11)

Consider the following two sets of Euclidean vectors:

\[U=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} -3 \, y z = -x \right\} \hspace{2em} W=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} -x - 3 \, z = -y \right\}.\]

Explain why one of these sets is a subspace of \(\mathbb{R}^3\), and why the other is not.

Answer.

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


Example 12

V4 - Subspaces (ver. 12)

Consider the following two sets of Euclidean vectors:

\[U=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} -5 \, x - 3 \, y - 5 \, z = 0 \right\} \hspace{2em} W=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} -3 \, {\left| z \right|} = -5 \, x - 5 \, y \right\}.\]

Explain why one of these sets is a subspace of \(\mathbb{R}^3\), and why the other is not.

Answer.

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


Example 13

V4 - Subspaces (ver. 13)

Consider the following two sets of Euclidean vectors:

\[U=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} -5 \, x = -2 \, y - 4 \, z \right\} \hspace{2em} W=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} x^{2} - 5 \, z = -4 \, y \right\}.\]

Explain why one of these sets is a subspace of \(\mathbb{R}^3\), and why the other is not.

Answer.

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


Example 14

V4 - Subspaces (ver. 14)

Consider the following two sets of Euclidean vectors:

\[U=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} -x + 2 \, y = -3 \, z \right\} \hspace{2em} W=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} -x z = 2 \, y \right\}.\]

Explain why one of these sets is a subspace of \(\mathbb{R}^3\), and why the other is not.

Answer.

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


Example 15

V4 - Subspaces (ver. 15)

Consider the following two sets of Euclidean vectors:

\[U=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} -4 \, y + 4 \, z = x \right\} \hspace{2em} W=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} z^{3} + 4 \, y = -4 \, x \right\}.\]

Explain why one of these sets is a subspace of \(\mathbb{R}^3\), and why the other is not.

Answer.

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


Example 16

V4 - Subspaces (ver. 16)

Consider the following two sets of Euclidean vectors:

\[U=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} -4 \, y - 3 \, z = -5 \, x \right\} \hspace{2em} W=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} z^{2} - 3 \, y = -4 \, x \right\}.\]

Explain why one of these sets is a subspace of \(\mathbb{R}^3\), and why the other is not.

Answer.

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


Example 17

V4 - Subspaces (ver. 17)

Consider the following two sets of Euclidean vectors:

\[U=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} -5 \, x - 3 \, y = -4 \, z \right\} \hspace{2em} W=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} -3 \, y - 4 \, z - 5 \, {\left| x \right|} = 0 \right\}.\]

Explain why one of these sets is a subspace of \(\mathbb{R}^3\), and why the other is not.

Answer.

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


Example 18

V4 - Subspaces (ver. 18)

Consider the following two sets of Euclidean vectors:

\[U=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} 3 \, x + 2 \, y + 2 \, z = 0 \right\} \hspace{2em} W=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} 2 \, y z + 2 \, x = 0 \right\}.\]

Explain why one of these sets is a subspace of \(\mathbb{R}^3\), and why the other is not.

Answer.

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


Example 19

V4 - Subspaces (ver. 19)

Consider the following two sets of Euclidean vectors:

\[U=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} -5 \, z = 3 \, x - 2 \, y \right\} \hspace{2em} W=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} -2 \, y - 5 \, z + 3 \, {\left| x \right|} = 0 \right\}.\]

Explain why one of these sets is a subspace of \(\mathbb{R}^3\), and why the other is not.

Answer.

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


Example 20

V4 - Subspaces (ver. 20)

Consider the following two sets of Euclidean vectors:

\[U=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} 4 \, x - 4 \, y + z = 0 \right\} \hspace{2em} W=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} -4 \, {\left| z \right|} = 4 \, x + y \right\}.\]

Explain why one of these sets is a subspace of \(\mathbb{R}^3\), and why the other is not.

Answer.

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


Example 21

V4 - Subspaces (ver. 21)

Consider the following two sets of Euclidean vectors:

\[U=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} -z = -4 \, x + 2 \, y \right\} \hspace{2em} W=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} -x z - 4 \, y = 0 \right\}.\]

Explain why one of these sets is a subspace of \(\mathbb{R}^3\), and why the other is not.

Answer.

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


Example 22

V4 - Subspaces (ver. 22)

Consider the following two sets of Euclidean vectors:

\[U=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} x^{3} - 3 \, y - z = 0 \right\} \hspace{2em} W=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} -x - 3 \, y + 5 \, z = 0 \right\}.\]

Explain why one of these sets is a subspace of \(\mathbb{R}^3\), and why the other is not.

Answer.

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


Example 23

V4 - Subspaces (ver. 23)

Consider the following two sets of Euclidean vectors:

\[U=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} z^{3} - 2 \, x - y = 0 \right\} \hspace{2em} W=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} x - y - 2 \, z = 0 \right\}.\]

Explain why one of these sets is a subspace of \(\mathbb{R}^3\), and why the other is not.

Answer.

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


Example 24

V4 - Subspaces (ver. 24)

Consider the following two sets of Euclidean vectors:

\[U=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} -5 \, x + 2 \, y - 3 \, z = 0 \right\} \hspace{2em} W=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} 2 \, y - 5 \, z - 3 \, {\left| x \right|} = 0 \right\}.\]

Explain why one of these sets is a subspace of \(\mathbb{R}^3\), and why the other is not.

Answer.

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


Example 25

V4 - Subspaces (ver. 25)

Consider the following two sets of Euclidean vectors:

\[U=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} y^{2} - 4 \, z = 4 \, x \right\} \hspace{2em} W=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} -4 \, y = -5 \, x + 4 \, z \right\}.\]

Explain why one of these sets is a subspace of \(\mathbb{R}^3\), and why the other is not.

Answer.

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


Example 26

V4 - Subspaces (ver. 26)

Consider the following two sets of Euclidean vectors:

\[U=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} y^{3} + 4 \, x - 4 \, z = 0 \right\} \hspace{2em} W=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} 4 \, y = -4 \, x + 5 \, z \right\}.\]

Explain why one of these sets is a subspace of \(\mathbb{R}^3\), and why the other is not.

Answer.

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


Example 27

V4 - Subspaces (ver. 27)

Consider the following two sets of Euclidean vectors:

\[U=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} z^{2} + 2 \, x - 5 \, y = 0 \right\} \hspace{2em} W=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} -5 \, z = -4 \, x + 2 \, y \right\}.\]

Explain why one of these sets is a subspace of \(\mathbb{R}^3\), and why the other is not.

Answer.

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


Example 28

V4 - Subspaces (ver. 28)

Consider the following two sets of Euclidean vectors:

\[U=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} -4 \, y - 5 \, z + 2 \, {\left| x \right|} = 0 \right\} \hspace{2em} W=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} 2 \, x - 5 \, z = -4 \, y \right\}.\]

Explain why one of these sets is a subspace of \(\mathbb{R}^3\), and why the other is not.

Answer.

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


Example 29

V4 - Subspaces (ver. 29)

Consider the following two sets of Euclidean vectors:

\[U=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} 3 \, x + 3 \, y - z = 0 \right\} \hspace{2em} W=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} 3 \, y z = -x \right\}.\]

Explain why one of these sets is a subspace of \(\mathbb{R}^3\), and why the other is not.

Answer.

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


Example 30

V4 - Subspaces (ver. 30)

Consider the following two sets of Euclidean vectors:

\[U=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} z^{3} - 4 \, x - 4 \, y = 0 \right\} \hspace{2em} W=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} -4 \, y - 4 \, z = x \right\}.\]

Explain why one of these sets is a subspace of \(\mathbb{R}^3\), and why the other is not.

Answer.

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


Example 31

V4 - Subspaces (ver. 31)

Consider the following two sets of Euclidean vectors:

\[U=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} x - 4 \, z = 3 \, y \right\} \hspace{2em} W=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} 3 \, y - 4 \, z + {\left| x \right|} = 0 \right\}.\]

Explain why one of these sets is a subspace of \(\mathbb{R}^3\), and why the other is not.

Answer.

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


Example 32

V4 - Subspaces (ver. 32)

Consider the following two sets of Euclidean vectors:

\[U=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} 3 \, y z = 5 \, x \right\} \hspace{2em} W=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} 3 \, y = 5 \, x - 5 \, z \right\}.\]

Explain why one of these sets is a subspace of \(\mathbb{R}^3\), and why the other is not.

Answer.

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


Example 33

V4 - Subspaces (ver. 33)

Consider the following two sets of Euclidean vectors:

\[U=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} -3 \, x y + 4 \, z = 0 \right\} \hspace{2em} W=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} -3 \, x + 4 \, y = -4 \, z \right\}.\]

Explain why one of these sets is a subspace of \(\mathbb{R}^3\), and why the other is not.

Answer.

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


Example 34

V4 - Subspaces (ver. 34)

Consider the following two sets of Euclidean vectors:

\[U=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} x^{2} + 5 \, y = -2 \, z \right\} \hspace{2em} W=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} 5 \, x = -2 \, y + 5 \, z \right\}.\]

Explain why one of these sets is a subspace of \(\mathbb{R}^3\), and why the other is not.

Answer.

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


Example 35

V4 - Subspaces (ver. 35)

Consider the following two sets of Euclidean vectors:

\[U=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} x^{3} + 5 \, y - 5 \, z = 0 \right\} \hspace{2em} W=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} 5 \, x + 5 \, y - 5 \, z = 0 \right\}.\]

Explain why one of these sets is a subspace of \(\mathbb{R}^3\), and why the other is not.

Answer.

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


Example 36

V4 - Subspaces (ver. 36)

Consider the following two sets of Euclidean vectors:

\[U=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} 2 \, x + 2 \, y - z = 0 \right\} \hspace{2em} W=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} 2 \, {\left| z \right|} = -x + 2 \, y \right\}.\]

Explain why one of these sets is a subspace of \(\mathbb{R}^3\), and why the other is not.

Answer.

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


Example 37

V4 - Subspaces (ver. 37)

Consider the following two sets of Euclidean vectors:

\[U=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} 4 \, y z = 2 \, x \right\} \hspace{2em} W=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} 4 \, z = 2 \, x - 3 \, y \right\}.\]

Explain why one of these sets is a subspace of \(\mathbb{R}^3\), and why the other is not.

Answer.

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


Example 38

V4 - Subspaces (ver. 38)

Consider the following two sets of Euclidean vectors:

\[U=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} -y = -5 \, x - 4 \, z \right\} \hspace{2em} W=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} y^{2} - 4 \, x - z = 0 \right\}.\]

Explain why one of these sets is a subspace of \(\mathbb{R}^3\), and why the other is not.

Answer.

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


Example 39

V4 - Subspaces (ver. 39)

Consider the following two sets of Euclidean vectors:

\[U=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} 2 \, y = -2 \, x + 2 \, z \right\} \hspace{2em} W=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} 2 \, {\left| x \right|} = 2 \, y - 2 \, z \right\}.\]

Explain why one of these sets is a subspace of \(\mathbb{R}^3\), and why the other is not.

Answer.

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


Example 40

V4 - Subspaces (ver. 40)

Consider the following two sets of Euclidean vectors:

\[U=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} 4 \, x - 2 \, y - z = 0 \right\} \hspace{2em} W=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} -2 \, x z = -y \right\}.\]

Explain why one of these sets is a subspace of \(\mathbb{R}^3\), and why the other is not.

Answer.

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


Example 41

V4 - Subspaces (ver. 41)

Consider the following two sets of Euclidean vectors:

\[U=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} -5 \, x = 2 \, y - 5 \, z \right\} \hspace{2em} W=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} x^{3} - 5 \, y = 2 \, z \right\}.\]

Explain why one of these sets is a subspace of \(\mathbb{R}^3\), and why the other is not.

Answer.

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


Example 42

V4 - Subspaces (ver. 42)

Consider the following two sets of Euclidean vectors:

\[U=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} -5 \, x y = 4 \, z \right\} \hspace{2em} W=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} -2 \, x + 4 \, y - 5 \, z = 0 \right\}.\]

Explain why one of these sets is a subspace of \(\mathbb{R}^3\), and why the other is not.

Answer.

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


Example 43

V4 - Subspaces (ver. 43)

Consider the following two sets of Euclidean vectors:

\[U=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} -4 \, x - 4 \, y = 5 \, z \right\} \hspace{2em} W=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} -4 \, x y - 4 \, z = 0 \right\}.\]

Explain why one of these sets is a subspace of \(\mathbb{R}^3\), and why the other is not.

Answer.

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


Example 44

V4 - Subspaces (ver. 44)

Consider the following two sets of Euclidean vectors:

\[U=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} -x y = -z \right\} \hspace{2em} W=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} -y = 2 \, x - z \right\}.\]

Explain why one of these sets is a subspace of \(\mathbb{R}^3\), and why the other is not.

Answer.

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


Example 45

V4 - Subspaces (ver. 45)

Consider the following two sets of Euclidean vectors:

\[U=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} 3 \, x = -4 \, y - 2 \, z \right\} \hspace{2em} W=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} x^{2} + 3 \, y = -4 \, z \right\}.\]

Explain why one of these sets is a subspace of \(\mathbb{R}^3\), and why the other is not.

Answer.

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


Example 46

V4 - Subspaces (ver. 46)

Consider the following two sets of Euclidean vectors:

\[U=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} -4 \, x - 2 \, y + z = 0 \right\} \hspace{2em} W=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} -4 \, y z = -2 \, x \right\}.\]

Explain why one of these sets is a subspace of \(\mathbb{R}^3\), and why the other is not.

Answer.

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


Example 47

V4 - Subspaces (ver. 47)

Consider the following two sets of Euclidean vectors:

\[U=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} 2 \, y = -2 \, x - 2 \, z \right\} \hspace{2em} W=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} 2 \, y z = -2 \, x \right\}.\]

Explain why one of these sets is a subspace of \(\mathbb{R}^3\), and why the other is not.

Answer.

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


Example 48

V4 - Subspaces (ver. 48)

Consider the following two sets of Euclidean vectors:

\[U=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} -3 \, y = 4 \, x + 5 \, z \right\} \hspace{2em} W=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} 4 \, x - 3 \, y + 5 \, {\left| z \right|} = 0 \right\}.\]

Explain why one of these sets is a subspace of \(\mathbb{R}^3\), and why the other is not.

Answer.

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


Example 49

V4 - Subspaces (ver. 49)

Consider the following two sets of Euclidean vectors:

\[U=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} z^{3} - 3 \, x - 2 \, y = 0 \right\} \hspace{2em} W=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} -2 \, z = -5 \, x - 3 \, y \right\}.\]

Explain why one of these sets is a subspace of \(\mathbb{R}^3\), and why the other is not.

Answer.

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


Example 50

V4 - Subspaces (ver. 50)

Consider the following two sets of Euclidean vectors:

\[U=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} x + 4 \, z = -2 \, y \right\} \hspace{2em} W=\left\{ \left[\begin{array}{r} x \\ y \\ z \end{array}\right] \hspace{0.2em}\middle|\hspace{0.2em} x - 2 \, y + 4 \, {\left| z \right|} = 0 \right\}.\]

Explain why one of these sets is a subspace of \(\mathbb{R}^3\), and why the other is not.

Answer.

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