Linear combination

In linear algebra, linear combinations are the addition of scalar multiples of vectors. We introduce some very important definitions below.

• Span — for some set of vectors in ${\mathbb{R}}^n$, we define their span $\text{span}({\vec{u}}_1,\dots ,{\vec{u}}_m)$ as the set of all linear combinations of the vectors in the set.
  • How do we determine if a vector $\vec{b}$ is in the span? We can row reduce $A\vec{x}=\vec{b}$.
  • The span is the image of the linear transformation described by the set.
• Linear independence — we say a set of vectors is linearly independent if the solution to the equation $c_1{\vec{u}}_1+\cdots +c_m{\vec{u}}_m=\vec{0}$ is the trivial solution $c_1=c_2=\cdots =c_m=0$.