matrix proof

[lollipop2046]

I have to prove that if M is a square upper triangular matrix, with nonzero diagonal entries, then its columns of M are linearly independent.
Intuitively I know that they are independent but I can only put it into english but not seem to be able to show it mathematically. Can you guys give me some hint on how to start this problem?

 

[tkhunny]

The matrix, A, is nxn with elements a_i,j

b_i are constants - some nonzero

Hint:
0 ≠ a_1,1 = b₂a_2,1 + b₃a_3,1 + b₄a_4,1 + ... + b_na_n,1 = 0
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[pka]

Suppose that C<SUB>1</SUB>, C<SUB>2</SUB>,…, C<SUB>n</SUB> are the n columns of the matrix and b<SUB>1</SUB>C<SUB>1</SUB>+ b<SUB>2</SUB>C<SUB>2</SUB>+…+b<SUB>n</SUB>C<SUB>n</SUB>=0.
Now the only non-zero entry in the nth row is a<SUB>n,n</SUB> this implies that b<SUB>n</SUB> must be 0.
In the (n−1)th row we have b<SUB>n−1</SUB>a<SUB>n-1,n-1</SUB>+ b<SUB>n</SUB>a<SUB>n-1,n</SUB>=0.
But b<SUB>n</SUB>=0, therefore b<SUB>n-1</SUB> also must be 0 because a<SUB>n-1,n-1</SUB> is not 0.
You need to show that all the b<SUB>i</SUB> equal 0.