Vectors a and b are linearly independent. Show vectors (a+b) and (a-b) are linearly independent

Thanks

we have to show that if C1(a+b) + C2(a-b) = 0, then C1=C2=0

C1(a+b) + C2(a-b) = 0
==> (C1+C2)a + (C1-C2)b = 0
since a and b are linearly independent, C1+C2=0 and C1-C2=0

C1+C2=0
C1-C2=0
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2C1=0 ==> C1=0 ==> C2=0