Consider the list of n n-vectors:
1 1 1
0 1 1
a1= [ 0 ], a2= [ 0 ] ,, an= [ 1 ]
. . .
. . .
. . .
0 0 1

(The vector ai has its first i entries equal to one, and the remaining entries zero.)
Describe what happens when you run the Gram–Schmidt algorithm on this list of vectors, i. e., say what q1,..., qn are. Is a1,..., an a basis?