00:01
This question we are given a matrix a which is equals to 1, minus 1, 3, 2, 1, minus 4, 1, minus 2, 4, minus 2 and 8.
00:17
So this is the matrix given to us in the question.
00:20
We have to find row space and null space basis for the given matrix a.
00:33
So let us find out the row reduced echelon form for the given matrix a.
00:41
So we are given the matrix 1, minus 2, 0, 4, 1.
00:47
So here it is minus 1, 1, 1 and minus 2, 3, minus 4, minus 2 and 8.
00:55
So if we apply the row operations, so let the first row operation be row 2 becomes row 2 plus 2 times row 1.
01:04
The second row operation be row 4 becomes row 4 minus 4 times row 1.
01:12
So we get the resultant matrix as 1, minus 1, 3, 0, minus 1, 2, 0, 1, minus 2, 0, 2, minus 4.
01:28
So now if we apply the row operation that row 3 becomes row 3 plus row 2 and row 4 becomes row 4 plus 2 times row 2.
01:44
Then we get the resultant matrix as, so the resultant matrix will be 1, minus 1, 3, 0, minus 1, 2, 0, 0, 0 and here also we get 0, 0 and 0.
02:03
So we can see that there are two pivot positions at here and at this point.
02:11
So row 1 and row 2 form the basis.
02:16
So let us complete the row reduce echelon form.
02:20
So if we now take the row operation that row 1 becomes row 1 minus row 2, then we get the final matrix as 1, 0, 1, 0, minus 1, 2, 0, 0, 0 and here also we get the row entire complete of 0.
02:45
So now we have pivot at this position and pivot at this position.
02:49
So row 1 and row 2 form the basis.
02:53
So we have 1, 0, 1 and 0, minus 1, 2.
02:59
These form the basis of row space of a.
03:06
So this is the final answer for the row space of a.
03:12
Now let us find the null space of a...