Answer:
4( -1, y)T + 2( 3x, 10)T = (8, 24)T ( -4, 4y)T + ( 6x, 20)T = (8, 24)T (6x-4, 4y+20)T = (8, 24)T 6x-4 = 8 4y+20 = 24 6x = 12 x = 2 4y = 4 y = 1
4( -1, y)T + 2( 3x, 10)T = (8, 24)T ( -4, 4y)T + ( 6x, 20)T = (8, 24)T (6x-4, 4y+20)T = (8, 24)T 6x-4 = 8 4y+20 = 24 6x = 12 x = 2 4y = 4 y = 1
In the original equation the "=" sign means vector equality. Equating corresponding elements yields two equations where the "=" sign means scalar equality (or real number equality.)
By equating corresponding elements a column matrix equation changes into several scalar equations, one equation for each dimension of the column matrix. Then, sometimes, the scalar equations can be solved for the unknowns. But this does not always work.
Here is a case where it does not work: Find x and y :
4(x, 5)T + 2(y, 1)T = 2(12, 11)T
This looks OK, but doing some work reveals a problem:
4( x, 5)T + 2( y, 1)T = 2(12, 11)T
(4x, 20)T + (2y, 2)T = (24, 22)T
(4x+2y, 22 )T = (24, 22)T
4x + 2y = 24
22 = 22
The two scalar equations you get by equating corresponding elements do not contain enough information to proceed.
Find a and x :
a( -1, 5 )T + 2( 3x, 10 )T = ( 8, 25 )T
(Before you start fiddling, take a guess: can it be done?)