If vector group A1, A2, A3 are linearly independent, it is proved that vector group B = a1 + 2A2, B2 = A2 + 2A3, B3 = A3 + 2A1 are linearly independent

If vector group A1, A2, A3 are linearly independent, it is proved that vector group B = a1 + 2A2, B2 = A2 + 2A3, B3 = A3 + 2A1 are linearly independent

Let K1 (a1 + 2A2) + K2 (A2 + 2A3) + K3 (A3 + 2A1) = 0, that is to say, K1 = K2 = K3 = 0 (K1 + 2K3) a1 + (2K1 + K2) A2 + (2k2 + K3) A3 = 0, because vector group A1, A2, A3 are linearly independent, so K1 + 2K3 = 02k1 + K2 = 02k2 + K3 = 0, the solution is K1 = K2 = K3 = 0, so vector group B = a1 + 2A2, B2 = A2 + 2A3, B3 = A3 + 2A1 is linearly independent

Let B1 = a1 + 2A2, B2 = A2 + 2A3, B3 = A3 + 2A1, B4 = a1 + A2 + a3, prove that vector group B1, B2, B3, B4 are linearly correlated

Because B4 = 1 / 3 * B1 + 1 / 3 * B2 + 1 / 3 * B3,
So B4 can be expressed linearly by B1, B2 and B3,
Therefore, B 1, B 2, B 3, B 4 were linear correlation

Solve the equation A3 + B3 + ab-a2-b2 = 0
The result seems to be a + B = 1

If a > 0, b > 0, A3 + B3 = 2, a + B ≤ 2, ab ≤ 1

(a + b) 3-23 = A3 + B3 + 3a2b + 3ab2-8 = 3a2b + 3ab2-6 ∵ A3 + B3 = 2 {6 = 3 × 2 = 3 (A3 + B3) ∵ a + b) 3-23 = 3 (A2B + ab2-a3-b3) = 3 [AB (a + b) - (A3 + B3)] and ∵ A3 + B3 = (a + b) (A2 AB + B2) ∵ (a + b) 3-23 = 3 (a + b) [ab - (A2 AB + B2)] = 3 (a + B