Assume A = [[5,-2],[-6,-7]] and b = [[1],[2]] Write b as a linear combination of the columns of A.   b = __ [[5],[-2]] + ____ [[-6],[-7]]



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oldnick's profile pic

Posted on (Answer #1)

`A=[[5,-2],[-6,-7]]`   `b=[[1],[2]]`

called x and y  coefficients of combination we have the system:

`5x-2y= 1`


A solution exists if the detrminant A `!=0`

Indeed  `det A =-47`

So we have to calculate the determinats:



`det [[5,1],[-6,2]]` `=16`

so that:

`x=(det[[1,-2],[2,-7]])/(det[[5,-2],[-6,-7]])` `=3/47`

`y=(det[[5,1],[-6,2]])/(det[[5,-2],[-6,-7]])` `=-16/47`


pramodpandey's profile pic

Posted on (Answer #2)


`b=lambda[[5],[-2]]+mu[[-6],[-7]]` ,  and `lambda,mu!=0`




Solving above system of equations for `lambda ,and mu .`  We have  `mu=-12/47,lambda=-5/47`


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