Linear Algebra

The system of linear equations that forces the subject into existence: elimination worked in full, the three outcomes it can reach, what it costs at scale, and the discovery that only the coefficients ever mattered.

The objects a coefficient column turns out to be: two operations and nothing else, then length, angle and perpendicularity all derived from a single product.

Every system of equations restated as one question, whether the target is a combination of the columns, and the property of those columns that decides which of the three outcomes occurs.

The flat sets that the two operations cannot escape, the smallest description of one, why coordinates in it are unique, and the proof that its size is a property of the space rather than of the description.

A map that respects addition and scaling is completely fixed by what it does to a basis, so four numbers hold an entire transformation of the plane.

Multiplication derived from composition rather than quoted, which explains the shape rule, the associativity, the failure of commutativity, and two non-zero matrices whose product is zero.

When a transformation can be undone: signed area as the definition of the determinant, $ad-bc$ derived from it, the multiplication rule for free, and the inverse when the determinant is not zero.

What a collapsing transformation destroys and what survives: column space and null space, why their dimensions add to the number of columns, and the complete solution of a system as one answer plus a subspace.

What to do when the target is unreachable, which is the normal case with real data: projection onto a subspace, the normal equations derived from a single right angle, and a fitted line computed to the last residual.

The same transformation described in someone else's coordinates, why that description is $P^{-1}AP$, what survives the translation, and a matrix that turns out to be diagonal when the coordinates are chosen well.

The directions a transformation leaves on their own line, found by asking when $A-\lambda I$ collapses, with the rotation that has none and the shear that has too few.

Eigenvectors used as a basis turn a matrix into a list of numbers, which makes repeated application arithmetic: a Markov chain settling to its steady state and a closed form for the Fibonacci numbers.

The family that always diagonalises, and with perpendicular axes: real eigenvalues proved, quadratic forms as ellipses, definiteness as a second derivative test, and where the symmetric matrices in applied work come from.