Vector Subspace: Show S is a Subspace, Determine Basis & Find Dimension

[quanta13]

Homework Statement

Can you help please? I have this problem:
Let S be the set of all vectors parallel to the hyper-plane 4x +2y+z + 3r =0 in R^4 .
(a) Show that S is a subspace, (b) Determine a basis for S , (c) Find its dimension

Homework Equations

The Attempt at a Solution

S= { u=(x, y,z,r) | 4x+2y+z+3r=0} u is in R^4.

 

[Mark44]

Homework Statement

Can you help please? I have this problem:
Let S be the set of all vectors parallel to the hyper-plane 4x +2y+z + 3r =0 in R^4 .
(a) Show that S is a subspace, (b) Determine a basis for S , (c) Find its dimension

Homework Equations

The Attempt at a Solution

S= { u=(x, y,z,r) | 4x+2y+z+3r=0} u is in R^4.

That won't work. If the vectors in S are parallel to the hyperplane, then all of these vectors are perpendicular to a normal to the hyperplane. If you're given the equation of a hyperplane, do you know how to find a normal to this hyperplane?

 

[Fredrik]

If u is a vector and S is a hyperplane, what does it mean to say that u is parallel to S? You will need a definition of "parallel" to answer that.