Linear Algebra II

September 18, 2014

Lecture 9

Filed under: 2014 Fall — Y.K. Lau @ 9:31 PM

We give a brief summary of the vector space theory covered today.

Next, we recap the key ideas in the proof of the dimension formula:

${\dim (U+W)=\dim U+\dim W-\dim (U\cap W)}$.

If ${\{x_1,\cdots, x_d\}}$ is a basis for ${U\cap W}$, ${\{x_1,\cdots, x_d, u_1,\cdots, u_m\}}$ and ${\{x_1,\cdots, x_d, w_1,\cdots, w_p\}}$ are bases for ${U}$ and ${W}$ resp., then ${\{x_1,\cdots, x_d, u_1,\cdots, u_m, w_1,\cdots, w_p\}}$ is a basis for ${U+W}$.

To show the linear independence, we consider

$\displaystyle r_1x_1+\cdots+r_d x_d +s_1 u_1+\cdots+ s_mu_m$ ${+}$ $\displaystyle t_1 w_1+\cdots+t_p w_p$ ${=\underline{0}.}$

From this we deduce

$\displaystyle r_1x_1+\cdots+r_d x_d + s_1 u_1+\cdots+ s_mu_m$ $\displaystyle =$ $\displaystyle (-t_1)w_1+\cdots +(-t_p)w_p$           ${(*)}$

Keypoints:

1. The LHS is in ${U}$ while the RHS is ${W}$.
Thus ${(-t_1)w_1+\cdots +(-t_p)w_p\in W}$ ${\cap}$ ${U}$.
2. As ${\{x_1,\cdots, x_d\}}$ is a basis for ${U\cap W}$,
${(-t_1)w_1+\cdots +(-t_p)w_p = a_1x_1+\cdots a_dx_d}$.
Rearranging, we get

$\displaystyle a_1x_1+\cdots a_dx_d+t_1w_1+\cdots +t_pw_p =\underline{0}.$

3. As ${\{x_1,\cdots, x_d, w_1,\cdots, w_p\}}$ is a basis for ${W}$, we infer ${a_1=\cdots =a_d=t_1=\cdots =t_p=0}$.
4. Put ${t_1=\cdots =t_p=0}$ into ${(*)}$, ${r_1x_1+\cdots+r_d x_d+s_1 u_1+\cdots+ s_mu_m=\underline{0}}$.
5. As ${\{x_1,\cdots, x_d, u_1,\cdots, u_m\}}$ is a basis for ${U}$, we infer ${a_1=\cdots =a_d=s_1=\cdots =s_m=0}$.