SOLUTION: Let A: [v1 v2 v3] with v1=(1,-1,0)T, v2=(2,0,-2)T and v3=(3, -3,3)T a) Show that {v1,v2,v3 } is a basis for Col(A). b) Find an orthogonal basis for V=col(A) (Gram-Schmidt)

Question 1061962: Let A: [v1 v2 v3] with v1=(1,-1,0)T, v2=(2,0,-2)T and v3=(3, -3,3)T
a) Show that {v1,v2,v3 } is a basis for Col(A).
b) Find an orthogonal basis for V=col(A) (Gram-Schmidt)

Answer by rothauserc(4718) (Show Source): You can put this solution on YOUR website!
A is
:
1 2 3
-1 0 -3
0 -2 3
:
column vectors of A are
:
1 -1 0
2 0 -2
3 -3 3
:
the row reduced echelon form of col(A) is
:
1 0 0
0 1 0
0 0 1
:
therefore {v1, v2, v3} is a basis for col(A) because col(A) is invertible and the columns of col(A) form a basis of R^3
:
orthogonal basis for col(A) using Gram-Schmidt
:
first vector (1/square root(2), -1/square root(2), 0)
second vector (1/square root(6), 1/square root(6), -2/3)
third vector (1/square root(3), 1/square root(3), 1/square root(3))
:

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