If a vector Y3 lies in the plane determined by the two
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If a vector Y3 lies in the plane determined by the two vectors Y1 and Y2, then we can write Y3 as a linear combination of Y1 and Y2. That is, Y3 = k1Y1 + k2Y2 for some constants k1 and k2. But then k1Y1 + k2Y2 Y3 = (0, 0, 0). Show that if k1Y1 + k2Y2 + k3Y3 = (0, 0, 0), with not all of k1, k2, and k3 = 0, then the vectors are not linearly independent. [Hint: Start by assuming that k3 = 0 and show that Y3 is in the plane determined by Y1 and Y2. Then treat the other cases.] Note that this computation leads to the theorem that three vectors Y1, Y2, and Y3 are linearly independent if and only if the only solution of k1Y1 + k2Y2 + k3Y3 = (0, 0, 0) is k1 = k2 = k3 = 0.
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