Question

Check whether W is a subspace. If yes, give a base and the dimension of W. Give - as far as possible - a geometric interpretation of W.

W={(x 1,x 2​ ,0)ᵀ :x1,x 2​ ∈R}​

Answer 1

W is a subspace with a base of {(1,1,0)} and dimension 1, representing a plane parallel to the xy-plane and passing through the origin.

Step-by-step explanation:

To determine whether W is a subspace, we need to check if it satisfies three conditions: closure under addition, closure under scalar multiplication, and contains the zero vector. Given W={(x1,x2,0)ᵀ : x1,x2∈R}, we can see that it satisfies all three conditions.

To find a base for W, we can set x1=x2=1, which gives us (1,1,0). Therefore, {(1,1,0)} is a base for W.

The dimension of W is 1, since it only requires one vector to span the space.

Geometrically, W represents a plane in three-dimensional space. It is a plane parallel to the xy-plane and passing through the origin.