00:01
In this question, we are asked to find the basis for the null space of the matrix a and then find the nullity of the matrix a.
00:08
To do that, we need to recall what is the null space.
00:12
Null space of the matrix a consists of all vectors x, for which ax equals to 0.
00:20
Therefore, we need to find basis for the set of solutions of ax equals 0.
00:25
To do that, we need first to find all solutions of the system.
00:29
The augmented matrix for the system is 1 -0 -negative 3.
00:33
1 negative 1 010, 010300, and 00001190, where 0 -0 -0 is the right -hand side.
00:51
Now, this augmented matrix is equivalent to the system of equations.
00:56
X1 minus 3x3 plus x4 minus x5 equals 0, x2, x5 equals 0, x2, x2, plus 3 x4 equals 0 and x4 plus 9 x5 equals 0.
01:22
Now we want to write down the solution set in terms of the main variables and free variables.
01:31
The main variables are the variables corresponding to the pivot columns and the three variables are the variables corresponding to the non -pivot columns...