2 8 points A Special Subspace 3 5 n matrix and that k is a

2. (8 points) A Special Subspace 3, 5 × n matrix and that k is a constant real nuinber. Suppose that A is an n Define S as the set o e S as the set of all vectors x that satisfy the equation Ax = kx. (a) Use the definition of a subspace to show that the set S forms a subspace of R\"

Solution

Given an n*n matrix A and a real constant k.

S is a subset of Rⁿ. If for vector a vector x in Rⁿ , Ax=kx then x belongs to S.

First, we show that 0 vector belongs to S.

A0=0=k0 implies that 0 belongs to S.

Second, we show that if x and y are in S, then x+y will also belong to it.

As x & y are in S, therefore Ax=kx & Ay=ky.   

Which gives us, A(x+y)=Ax+Ay=kx+ky=k(x+y) which implies x+y belongs to S.

Third, we show that for any constant c from R and for any vector x form S, cx belongs to S.

As x belongs to S => Ax=kx

So, A(cx)= cAx=ckx=kcx=k(cx) and this holds for any cin R & x in S.

Therefore, we can conclude that S is a subspace of Rⁿ.

I hope you understand the process, if yes, hit the thumbs up.