Question

How can i prove that the solution set of a homogeneous linear system in n unknowns is a subspace of R^n?

Answer 1

For a system to be a subspace, it should be closed under addition and scalar multiplication.

Answer 2

You need to show 3 things. Let your system be represented by the matrix M. If \(A=\{x\mid Mx=0\}\), then:

1. \(A\ne \emptyset.\)
2. \(x,y\in A\Longrightarrow x+y\in A\) (Closure under addition)
3. \(c\in \mathbb R,x\in A\Longrightarrow cx\in A\) (Closure under scalar multiplication.)

For the first one, can you think of a vector that when you calculate M times that vector it will always come out to zero?

For the second, if \(Mx=0\) and \(My=0\), then \(M(x+y)=?\)

For the third, if \(Mx=0\), and c is a real number, then \(M(cx)=?\)