Students will learn what it means for a subset of \(\mathbb{R}^n\) to be a subspace. That is, if \(V\) is a subset of \(\mathbb{R}^n,\) then \(V\) is a subspace if and only if \(V\) contains the zero vector, is closed under scalar multiplication, and is closed under addition. As an example, students will see that \(V = \{0\}\) is a subspace of \(\mathbb{R}^n.\) Then students will be asked to determine whether various sets are subspaces. Here are a few examples:

Is the following set \(S\) a subspace of \(\mathbb{R}^2?\)

$$S = \left\{\begin{bmatrix}x_1 \\ x_2\end{bmatrix} \in \mathbb{R}^2 : x_1 \ge 0\right\}$$Is the following set \(T\) a subspace of \(\mathbb{R}^n?\)

$$T = \text{span}(\vec{v_1}, \vec{v_2}, \vec{v_3})$$Is the following set \(U\) a subspace of \(\mathbb{R}^n?\)

$$U = \text{span}\left(\begin{bmatrix}1 \\ 1\end{bmatrix}\right)$$Here's a lesson, and the answer to each of the aforementioned problems.