Draw the following subsets of $\mathbb{R}^n$

Key concept: This is a set-description and geometric-interpretation problem from linear algebra. The goal is to translate the coordinate condition into a geometric object in the plane.

Step by step

For

$\{\mathbf{x}=(x_1,x_2)\in\mathbb{R}^2\mid x_1=1\},$

the first coordinate is fixed at 1 while the second coordinate can be any real number.

Geometric meaning

This is the vertical line

$x=1$

in the plane.

How to draw it

• Mark the point $(1,0)$ on the $x$-axis.
• Draw a straight line parallel to the $y$-axis through that point.
• Extend it infinitely upward and downward.

Set interpretation

Every point on the line has the form

$(1,t),\quad t\in\mathbb{R}.$

Pitfall alert

Do not confuse $x_1=1$ with a single point. The second coordinate is free, so the set is not just $(1,0)$ or any other isolated point. Also, in $\mathbb{R}^2$, the condition $x_1=1$ always gives a vertical line, not a horizontal one.

Try different conditions

If the condition were $x_2=1$, the graph would be the horizontal line $y=1$. In higher dimensions, a condition like $x_1=1$ in $\mathbb{R}^n$ describes a hyperplane, not just a line.

Further reading

subset of R^n, hyperplane, coordinate condition