Question

Let V=ℝ2 and let H be the subset of V of all points on the line 4x+3y=12. Is H a subspace of the vector space V?

Answer 1

No, it isn't.

Explanation:

We have
`V=IR^(2)` and let
`H` be the subset of
`V` of all points on the line

`4x+3y=12`

We need to find if
`H` is a subspace of the vector space
`V`.

In
`IR^(2)` all the possibilities for own subspace of the vector space
`IR^(2)` are :

• 
  `IR^(2)` itself.

• The vector
  `0_(IR^(2))=\left[\begin{array}{c}0&0\end{array}\right]`

• All lines in
  `IR^(2)` that passes through the origin (
  `0_(IR^(2))=\left[\begin{array}{c}0&0\end{array}\right]` )

We know that
`H` is the subset of
`IR^(2)` of all points on the line
`4x+3y=12`

If we look at the equation, the point
`\left[\begin{array}{c}0&0\end{array}\right]` doesn't verify it because :

`4x+3y=12\\4(0)+3(0)=12\\0=12`

Which is an absurd. Therefore,
`H` doesn't contain the origin (and
`H` is a line in
`IR^(2)`). Finally, it can't be a vector space of
`V=IR^(2)`