A Line is a Subspace if and only if its $y$-Intercept is Zero

Vector Space Problems and Solutions

Problem 663

Let $\R^2$ be the $x$-$y$-plane. Then $\R^2$ is a vector space. A line $\ell \subset \mathbb{R}^2$ with slope $m$ and $y$-intercept $b$ is defined by
\[ \ell = \{ (x, y) \in \mathbb{R}^2 \mid y = mx + b \} .\]

Prove that $\ell$ is a subspace of $\mathbb{R}^2$ if and only if $b = 0$.

 
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Proof.

We must prove two statements. First we show that if $b \neq 0$, then $\ell$ is not a subspace. Then we show that if $b=0$, then $\ell$ is a subspace.

In order for $\ell$ to be a subspace, it must contain the zero element $(0, 0)$. Plugging this point into the defining equation yields $b=0$. Thus if $b \neq 0$, then $\ell$ cannot be a subspace.


Now we prove that if $b=0$, then $\ell$ is a subspace. We have already shown that $(0, 0) \in \ell$. Now suppose we have two points $(x_1, y_1) , (x_2 , y_2) \in \ell$. Then we have
\[y_1 + y_2 = m x_1 + m x_2 = m(x_1 + x_2)\] and so $(x_1 + x_2 , y_1 + y_2)$ is contained in $\ell$.


Finally for $c \in \mathbb{R}$ we must show that $c(x_1 , y_1) = ( cx_1 , c y_1 )$ lies in $\ell$. We check
\[c y_1 = c ( m x_1) = m (c x_1),\] and so $(c x_1 , c y_1 ) \in \ell$. This proves that if $b=0$, then $\ell$ is a subspace.


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