HW 4: If the set {v1, v2, v3, v4} is linearly independent, then {v1, v2, v3, v4, v1+v4} is linearly independent.
FALSE. The vector v1+v4 is a linear combination of v1 and v4. Thus, {v1, v2, v3, v4, v1+ v4} is linearly dependent.
To prove that {v1, v2, v3, v4, v1+v4} is linearly independent, we must show that the only solution to the equation:
a1v1 + a2v2 + a3v3 + a4v4 + a5(v1+v4) = 0
is the trivial solution (a1 = a2 = a3 = a4 = a5 = 0).
We can rewrite the equation above as:
(a1 + a5)v1 + a2v2 + a3v3 + (a4 + a5)v4 = 0
Since {v1, v2, v3, v4} is linearly independent, the only solution to the equation above is a1 + a5 = a2 = a3 = a4 + a5 = 0.
Using the third and first equations, we can isolate a4 and a5:
a4 = -a5
a1 = -a5
Substituting these back into the second equation, we get:
a2 = 0
And substituting into the fourth equation, we get:
a1 + a4 = 0
-a5 – a5 = 0
-2a5 = 0
a5 = 0
Thus, a1 = a2 = a3 = a4 = a5 = 0, which means {v1, v2, v3, v4, v1+v4} is linearly independent. Therefore, the statement is true.
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