1.4 Subspace

What is a Subspace

• we know from 1.2 Span if a span is a line it would be a line in the ${\mathbb{R}}^n$
  • and to get a subspace it has to span an area and has to be linearly independent from one another

  • a subspace can be thought of as a ‘line/plane’ that passes through the origin

    • intuitively: a subspace is a span of some vectors

• the formal definition

  • it is a set of vectors in ${\mathbb{R}}^n$ which is closed under linear combinations

    • (where if you take all possible linear combinations we get the same set of numbers)

• We can show that something isn’t a subspace using Counterexamples

Examples

Ex1

S=(\begin{bmatrix} 2t \\ t \end{bmatrix}|t\in\mathbb{R}) $$ - is a subspace of $\mathbb{R}^2$ because

let\begin{bmatrix}
2t_1 \
t_1
\end{bmatrix},\begin{bmatrix}
2t_2 \
t_2
\end{bmatrix}\in \mathbb{R}

- then #### Ex2

S=([x_{1},x_{2},x_{3}]|x_{1}+x_{2}+x_{3}=0)

- is a subspace of $\mathbb{R}^3$, because -