Is a Line Through the Origin Always a Subspace of R^n?

[hkus10]

1) Let x0 be a fixed vector in a vector space V. Show that the set W consisting of all scalar multiples cx0 of x0 is a subspace of V.

What techniques should I use to prove this?

2a) Show that a line lo through the origin of R^n is a subspace of R^n.
2b) show that a line l in R^n not passing through the origin is not a subspace of R^n.

What techniques and direction should I use to solve these problems?

Thanks

 

[Mark44]

1) Let x0 be a fixed vector in a vector space V. Show that the set W consisting of all scalar multiples cx0 of x0 is a subspace of V.

What techniques should I use to prove this?

Show that the 0 vector is in W.
Show that W is closed under vector addition. I.e., if w₁ and w₂ are in W, then so is w₁ + w₂.
Show that W is closed under scalar addition. I.e., if w is in W, then cw is also in W.

2a) Show that a line lo through the origin of R^n is a subspace of R^n.
2b) show that a line l in R^n not passing through the origin is not a subspace of R^n.

What techniques and direction should I use to solve these problems?

Thanks

Same ideas as in 1. For 2b, one or more of the conditions won't be satisfied.