Linear Combinations and Span

In Build, the combination

$$x_1\overline{u}_1 + x_2\overline{u}_2$$

is drawn tip to tail from the two sliders, $x_1\overline{u}_1$ first and then $x_2\overline{u}_2$ from the tip of $x_1\overline{u}_1$. Show numbers works out each vector scaled and the scaled vectors added. A scalar can also be typed as a fraction, such as 11/3.

In Target, a target $\overline{v}$ is added. After Show numbers, Next step reveals the hand solution one step at a time: one equation per component, then substitution or elimination, then the check. All steps shows or hides every step at once. The verdict and R’s answer appear with the last step; R’s answer is under Show R. After any change to $\overline{u}_1$, $\overline{u}_2$ or $\overline{v}$, the steps of the new problem start hidden. Under Matrix equation, the same problem is written as $A\overline{x} = \overline{v}$. The columns of $A$ are $\overline{u}_1$ and $\overline{u}_2$.

Along each faint line, $x_1$ or $x_2$ is a whole number. When $\overline{u}_1$ and $\overline{u}_2$ are not parallel, the grid covers the plane. When their tips lie on one line through the origin, the grid collapses onto that line. A target on the line is then a linear combination of $\overline{u}_1$ and $\overline{u}_2$ in infinitely many ways. A target off the line is not a linear combination of $\overline{u}_1$ and $\overline{u}_2$. Whether $\overline{v}$ lies on the line is noted on the board, and Show R gives the error that solve() returns.

Under More, in Board:

• Snap step. Dragged tips snap to whole numbers, halves or tenths.
• Combination grid and Tip to tail. Both drawings are on by default.

Keys: press 1 or 2 to pick a tip. In Target, press V to pick the target. Press an arrow key to move the picked tip one snap step. Hold Shift with an arrow key to move it five snap steps.