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

Answer:

No, it isn't.

Step-by-step 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 $V=IR^{2}$