show that the dimension of a subspace is less than or equal

show that the dimension of a subspace is less than or equal to the dimension of the vector space

Solution

To prove this we will show an example

Example: W={(c,2c,3c)|cR} is a subspace of R3. Now, yes, elements of W are 3-tuples, but this does not make W itself 3-dimensional.

Think of \"dimension\" as meaning the minimum number of parameters needed to describe the subspace, so for W this is \"1\". Notice that W consists of multiples of (1,2,3). This means that {(1,2,3)} is a basis for W and thus (since the basis has only 1 element), W is a 1-dimensional subspace of R3.