Linear Algebra Question

Question

the vector [-14,-7,19] is a linear combination of vectors [-3,-3,-3] and [10,9,-10] if and only if the matrix equation Ax = b has a solution x where  

 

A = [?  ?]    and b = [?]

      [?  ?]                 [?]

      [?  ?]                 [?]

Answer 1

If <-14, -7, 19> is a linear combination of <-3, -3, -3> and <10, 9, -10>, then 

 

<-14, -7, 19> = c₁<-3, -3, -3> + c₂<10, 9, -10>

 

So, -14 = -3c₁ + 10c₂

 

       -7 = -3c₁+ 9c₂

 

       19 = -3c₁ - 10c₂

 

The system of equations above can be written in matrix form as Ax = B, where:

 

        [ -3      10  ]               [ -14]            [c₁]

 A = [ -3        9  ]        B =  [ -7 ]       x = [c₂] 

        [ -3     -10  ]               [ 19 ]