Showing Linear Subspace & Affine Subspace of Vector a in R^n

[tk1234]

Let a be a fixed nonzero vector in R^n

a. Show that the set S of all vectors x such at ax=0 is a subspace of R^n.
b. show that if k is a nonzero real number, then the set A of all vectors x such that ax=k is an affine subspace of R^n, but not a linear subspace.how do i even show this.. I am so confused. help please! thanks.

 

[Dick]

Let a be a fixed nonzero vector in R^n

a. Show that the set S of all vectors x such at ax=0 is a subspace of R^n.
b. show that if k is a nonzero real number, then the set A of all vectors x such that ax=k is an affine subspace of R^n, but not a linear subspace.


how do i even show this.. I am so confused. help please! thanks.

Try?? Pretty please? Tell me what you might need to prove S is a subspace. You can look it up in the book if you want to. What's the definition of a subspace?

 

[tk1234]

i know that S needs to be closed under addition and numerical multiplication.. I am just not sure how to show this?!

 

[Dick]

i know that S needs to be closed under addition and numerical multiplication.. I am just not sure how to show this?!

If x1 is in S then a.x1=0. If x2 is in S then a.x2=0. (I'm using '.' for dot product). Can you give me some reason why that would mean x1+x2 is in S? What would you have show to prove x1+x2 is in S? That would give you closure under addition.