Linear algebra
1. Vectors, combinations, and measurement
What can a vector represent, reach, and measure?
Does moving an arrow change the vector?
Move the same arrow to a different starting point. Discover the difference between a position and a displacement.
Can we combine movements before making them?
Compare two paths, reverse their order, and scale one move. Discover the rules for vector addition and multiplication by a number.
How many controls does a vector need?
Build a four-component record one slot at a time. Discover how coordinate unit vectors turn numbers into directions.
Can direction stay fixed while length changes?
Stretch, reverse, and shrink a vector. Discover length, normalization, and the special case of the zero vector.
Why is the dot product connected to projection?
Predict how a shadow changes, then stretch the reference vector. Discover which changes the dot product notices.
Which targets can two vector controls reach?
Try to reach a target, then make the controls parallel. Discover why extra controls do not always create extra possibilities.
2. Constraints and solution families
When can equations be satisfied, and how much choice remains?
How many solutions can two equations have?
Move two lines from crossing to parallel to overlapping. Discover what it means to solve both equations at once.
Why can we subtract equations without losing the answer?
Turn a pair of equations into an easy answer, one reversible operation at a time. Watch the lines change while their intersection stays put.
What does a pivot actually tell us?
Read an echelon matrix before solving it. Discover how pivots distinguish determined variables, free choices, and contradictions.
What does a free variable let us choose?
Choose one coordinate and let the others follow. Build an entire family of solutions, then add a constraint that selects just one.
How are all solutions related to one solution?
Subtract two solutions and inspect the difference. Discover how one starting solution and zero-output changes describe a whole family.
Are equations and vector combinations the same problem?
Connect two constraints, an augmented matrix, column controls, and row measurements. Discover what Ax = b means.