Let A1, A2, A3 and A4 be four-dimensional vectors, and a = (A1, A2, A3 and A4) have known the general solution x = K (1,0,1,0) ^ t, and find the solution of A1, A2, A3 and A4 of vector group The answer is R (a) = 3. How did this come about? Finding the maximal independent group of A1, A2, A3, A4 of vector group

Let A1, A2, A3 and A4 be four-dimensional vectors, and a = (A1, A2, A3 and A4) have known the general solution x = K (1,0,1,0) ^ t, and find the solution of A1, A2, A3 and A4 of vector group The answer is R (a) = 3. How did this come about? Finding the maximal independent group of A1, A2, A3, A4 of vector group

Because there is only one vector in the general solution
So the fundamental solution system of AX = 0 contains one solution vector
So N-R (a) = 4-R (a) = 1
So r (a) = 3
And because (1,0,1,0) is the solution vector of AX = 0
So a1 + a3 = 0
So A1, A2, A4 are maximal independent groups of A1, A2, A3, A4

Let a = (A1, A2, A3, A4) and AI (I = 1,2,3,4) be 5-Dimensional vectors. If A2, A3, A4 are linearly independent and A4 = a1 + 2a2-a3, find the general solution of the system of equations AX = 0

Let the rank of vector group A1, A2, A3, A4 be 3, and the rank of vector group A1, A2, A3, A5 be 4, then the rank of vector group A1, A2, A3, a5-a4 is 4

Because the rank of A1, A2, A3, A5 is 4
So A1, A2, A3 are linearly independent, and A5 cannot be expressed linearly by A1, A2, A3
And because the rank of A1, A2, A3, A4 is 3
So A4 can be expressed linearly by A1, A2, A3
So a5-a4 can't be expressed linearly by A1, A2, A3
However, A1, A2, A3 are linearly independent
So A1, A2, A3, a5-a4 are linearly independent
So r (A1, A2, A3, a5-a4) = 4