3.5 Column Space

What is column space

• By writing vectors as columns and reducing to RREF, we can obtaina basis for Col(A) as a subset of the columns of A

  • If RREF of A is R, then the columns of A corresponding to the columns of R with leading 1s form a basis of Col(A)
    • (basicly RREF then only have the basis for the ones that are 1.4 Linear Independence)

• Ex

  • find a basis for the set of vectors as a subset of the vi vectors

$$\begin{array}{r}let,\overrightarrow{v_1}=\left[\begin{array}{ccc}1& 1& 1\end{array}\right],{\vec{v}}_2=\left[\begin{array}{ccc}3& 1& 4\end{array}\right],{\vec{v}}_3=\left[\begin{array}{ccc}3& -1& 5\end{array}\right],{\vec{v}}_4=\left[\begin{array}{ccc}2& 3& 1\end{array}\right]\\ A=[{\vec{v}}_1,{\vec{v}}_2,{\vec{v}}_3,{\vec{v}}_4]=\left[\begin{array}{cccc}1& 3& 3& 2\\ 1& 1& -1& 3\\ 1& 4& 5& 1\end{array}\right]\\ RREF(A)=\left[\begin{array}{cccc}1& 0& -3& 0\\ 0& 1& 2& 0\\ 0& 0& 0& 1\end{array}\right]\\ Basis(A)=\left[\begin{array}{ccc}1& 3& 2\\ 1& 1& 3\\ 1& 4& 1\end{array}\right]\end{array}$$