Linear Combination

A linear combination is in the form $a_1{v}_1+a_2{v}_2+\cdots +a_k{v}_k={\sum }_{i=1}^k{a}_i{v}_i$ Where $v_1,\dots ,v_n\in V,a_1,\dots ,a_n\in \mathbb{F}$

Closed Under Linear Combinations

If $S$ is a subspace and $\vec{u},\vec{v}\in S⟹a\vec{u}+b\vec{v}\in S$

This happens when a set $S$ is both:

• Closed Under Addition
• Closed Under Scalar Multiplication

Proving Individual Vectors from Linear Combination Example