Before you can even read the parent note, a whole toolbox of symbols is quietly assumed. This page builds every single one from zero, in an order where each idea rests only on the ones before it.
Figure s01 draws exactly this: the lavender arrow v=(3,2) runs from the origin (dark dot) to the coral point. Follow the dashed grey lines — "3 right" along the bottom, then "2 up" the side — to see how the two numbers spell out the arrow. The origin is where every vector on this whole page starts.
We need this because a linear transformation is a machine whose inputs and outputs are arrows. No arrows, nothing to transform.
Arrows can only be combined in two ways, and a linear map is defined entirely by respecting these two ways.
Figure s02 shows both moves. On the left, the lavender u and mint v are laid tip-to-tail, and the coral arrow u+v closes the path — that coral arrow is the sum. On the right, one lavender arrow w becomes butter-coloured 2w (twice as long, same direction) and coral −w (same length, flipped to point the opposite way). Those are the only two things you are ever allowed to do to arrows.
Why the topic needs these: the kernel is written kerT={v∈V:T(v)=0} — you literally cannot read that line without knowing ∈, the colon, and the braces. The parent note also insists "kerT⊆V" — that ⊆ is the whole point of a common mistake.
Figure s03 is that box: the lavender arrow "input v in V" feeds into the butter-coloured T-box, and the coral arrow "output T(v) in W" comes out the other side. The label T:V→W underneath names the whole picture. Keep this image in mind — the kernel and image are just questions about which arrows enter and leave this box.
Why the topic needs it: "linear transformation" is a special kind of function. You must first know what a function is before you can add the two fairness rules that make it linear.
Figure s04 builds the arrow (3,2) out of the basis: the short mint arrow e1 and short coral arrow e2 are the two building blocks. The dashed mint arrow is "3 steps of e1", the dashed coral arrow stacked on its tip is "2 steps of e2", and together they land exactly at the lavender (3,2). Every arrow in the plane is some recipe like this — that is what "a basis builds everything" means.
Why the topic needs these: the kernel and image are always subspaces, and rank/nullity are just their dimensions. See Rank–Nullity Theorem and Column space and null space for where these become the main event.
Why the topic needs it: the parent claims "image = column space" precisely because every output Ax is a scaling-and-adding of the columns. The mechanics of Ax live in Matrix multiplication. Together with Injective surjective bijective (the "one-to-one / onto" words) and Eigenvalues and eigenvectors (special arrows a map only stretches), these foundations complete the toolbox.
