Question

For the system

2x1 − 4x2 + x3 + x4 = 0,

x1 − 2x2 + 5x4 = 0,

find some vectors v1, . . . , vk such that the solution set to this system equals span(v1, . . . , vk).

Answer 1

The given homogeneos linear system of equations is

2x₁ − 4x₂ + x₃+ x₄ = 0,

x₁ − 2x₂ + 5x₄ = 0,

The coefficient matrix of this system is A(say) =

2	-4	1	1
1	-2	0	5

To solve the above system, we will reduce A to its RREF as under:

Multiply the 1st row by ½

Add -1 times the 1st row to the 2nd row

Multiply the 2nd row by -2

Add -1/2 times the 2nd row to the 1st row

Then the RREF of A is

1	-2	0	5
0	0	1	-9

This implies that the given linear system is equivalent to x₁ − 2x₂ + 5x₄ = 0 or, x₁ = 2x₂ - 5x₄ and x₃ -9x₄ = 0 or, x₃ =9x₄.

Then, (x₁,x₂,x₃,x₄)^T = (2x₂-5x₄ , x₂, 9x₄ , x₄)^T = x₂(2,1,0,0)^T+ x₄(-5, 0,9,1)^T = s(2,1,0,0)^T+ t(-5, 0,9,1)^T where s = x₂ and t = x₄ are arbitrary real numbers. This means that every solution to the given homogeneos linear system of equations is a linear combination of the vectors v₁ =(2,1,0,0)^T and v₂ = (-5, 0,9,1) .

Thus the solution set of the given homogeneos linear system of equations = span{v₁,v₂} = span{(2,1,0,0)^T , (-5, 0,9,1)^T }