1) Suppose that $\displaystyle \mathbf{v}$ is a vector in $\displaystyle \mathbb{R}^n$. Show that the line segment defined by:

$\displaystyle S=\{\mathbf{x}\in\mathbb{R}^n:\mathbf{x}=\lambda\m athbf{v}, \mbox{for} \ 0\leq\lambda\leq10\}$

is not a subspace of $\displaystyle \mathbb{R}^n$

Since its a line through the origin, the $\displaystyle \mathbf{0}$ vector is in $\displaystyle S$.

$\displaystyle \mathbf{x_1}=\lambda \mathbf{v_1} \ \ \mbox{and} \ \ \mathbf{x_2}=\lambda \mathbf{v_2}$ such that $\displaystyle \mathbf{x_1, x_2}\in \mathbf{x}$

$\displaystyle \mathbf{x_1}+\mathbf{x_2}=\lambda\mathbf{v_1}+\lam bda\mathbf{v_2}$

$\displaystyle =\mathbf{x_1}+\mathbf{x_2}=\lambda(\mathbf{v_1}+\m athbf{v_2})$

$\displaystyle \mathbf{v_1}+\mathbf{v_2}\in \mathbf{v}\Longrightarrow \mathbf{x_1}+\mathbf{x_2}\in \mathbf{x}$

Therefore, its closed under scalar multiplication

What would you do next for this question?

2) Show that the set:

$\displaystyle S=\{\mathbf{x}\in\mathbb{R}^n:2x_1+3x_2-4x_3=4x_1-2x_2+3x_3=0\}$

is a subspace of $\displaystyle \mathbb{R}^3$

For this question, do you only need to consider say $\displaystyle 2x_1+3x_2-4x_3=0$ and show that is a subspace?