Enlarge system to be basis

Question

We have $\{x_1,x_2\}$ vector system. The task wants from me to extend it to make basis of $V_4$.

$x_1$ = $(1,1,0,1)$ and $x_2$ = $(2,1,1,1)$

Answer 1

As a general method we can find two orthogonal vectors solving

• $(a,b,c,d) \cdot x_1=a+b+d=0$

• $(a,b,c,d) \cdot x_2=2a+b+c+d=0$

from here we obtain a solution with to free parameters $(s,t)$ and then

• $(s,t)=(1,0)\to x_3$
• $(s,t)=(0,1)\to x_4$

and since those are orthogonal to $x_1$ and $x_2$ they are linearly independent and form a basis.

Answer 2

You need to add two more linearly independent vectors to $(1,1,0,1),(2,1,1,1).$

You have many choices, for example $(1,0,0,0)$ and $(0,0,0,1)$ will do.

To check linear independence you find the determinant of the matrix $$ M = \begin{bmatrix}1&1&0&1\\2&1&1&1\\1&0&0&0\\0&0&0&1\end{bmatrix}$$

which is $1 \ne 0$

In general you find orthogonal vectors by solving a system of equations.

You want the dot product of the unknown vector with the known vectors to be $0$.