Conditions for Homogeneity/Straight Line

Theorem

The line $L = \alpha_1 x_1 + \alpha_2 x_2 = \beta$ is homogeneous if and only if $\beta = 0$.

Proof

Sufficient Condition

Let the line $L = \alpha_1 x_1 + \alpha_2 x_2 = \beta$ be homogeneous.

Then the origin $\tuple {0, 0}$ lies on $L$.

Hence:

\(\ds \alpha_1 0 + \alpha_2 0\)

\(=\)

\(\ds \beta\)

\(\ds \leadsto \ \ \)

\(\ds \beta\)

\(=\)

\(\ds 0\)

$\Box$

Necessary Condition

Let the equation of $L$ be $\alpha_1 x_1 + \alpha_2 x_2 = 0$.

Then:

$0 = \alpha_1 0 + \alpha_2 0 \in L$

and so:

$\tuple {0, 0}$ lies on $L$

Hence $L$ is homogeneous by definition.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 28$. Linear Transformations