Find he basis for each of the four fundamental subspaces ass

Find he basis for each of the four fundamental subspaces associated with this matrix: [2 6 5 15 3 9]

Solution

1. the 1 st and 2nd column [ 2 ,6 ] , [ 5 , 15 ] to generate these 2 vectors and the subspace spanned

by [ 1,3 ] as [2,6] = 2[ 1,3] , [5,15] = 5 [1,3] hence the basis is [ { 1,3} ]

2 . simalarly for th 2nd , 3rd column { (5,15), (3,9) } the basis is {(1,3) }

3 for the 3rd and 1st column also the same basis { (1,3)}

4 considering the 1 st, 2nd rows { ( 2,5,3) , ( 6,15 , 9) } the basis is { ( 2,5,3) } as the other vector is 3 times the 1st