Title: Default for Test JS
Course: JS sample testing course

Let \(W\) be the subset of \(\mathbb{R}^3\) made from the straight line through the origin and \( ([value], [value], [value]) \). Is \(W\) a subspace of \(\mathbb{R}^3\)?

True or false: \(W\) is a subspace?

If true write the subspace in parametric form: where the list of parameters \(\in \mathbb{R}\).

If false the give reasons below.

- True or false: \(W\) does not contain the origin (and so is not a subspace)?
- \(W\) is not a subspace because it is not closed under addition. For example, \(v_1 = \) and \(v_2 = \) are in \(W\) but their sum is not.
- \(W\) is not a subspace because it is not closed under scalar multiplication. For example, \(w_1 = \) is in \(W\) and \(\lambda = \) is a scalar but the scalar multiple \(\lambda w_1\) is not in \(W\).
Feedback: A line through the origin is a subspace. The subspace can be described in parametric form with a single parameter, multiplying the position vector of any non-zero point on the line.