1 · A vector is a list of numbers with a meaning
Write [1, 0] on paper. In quantum computing that list is the state "definitely 0", and [0, 1] is "definitely 1". A qubit's state is just a two-number list [α, β]: the amplitudes. All the quantum drama lives in ordinary lists of (complex) numbers.
2 · A basis is a choice of questions
The same vector can be described against different reference directions, different bases. In quantum mechanics, choosing a measurement basis is literally choosing which question to ask the qubit, and the same state answers different questions with different statistics. That's why "measure in X instead of Z" is a meaningful, physical act.
3 · A matrix is a transformation
A matrix is not a table of numbers; it is a machine that turns one vector into another: rotating it, flipping it, shearing it. Quantum gates are exactly this: small matrices applied to the state list. The X gate is the matrix that swaps [α, β] to [β, α]; the Hadamard gate is the matrix that mixes them into equal blends. A circuit is just matrix after matrix.
Achtung, Analogie: „Maschine" unterschätzt eine Eigenschaft: Gate-Matrizen sind umkehrbar (unitär). Bei der Zustandstransformation geht nichts verloren – genau deshalb spielt die Messung, die tatsächlich verlustbehaftet ist, eine so besondere Rolle.
4 · Tensor products: how systems combine
One qubit is a 2-number list. Two qubits are not two separate 2-number lists. They are one 4-number list ([00, 01, 10, 11] amplitudes), built by the tensor product. Three qubits: 8 numbers. Fifty qubits: more numbers than a supercomputer can comfortably hold. This single fact is where the exponential state space comes from, and entanglement is what happens when that joint list cannot be factored back into separate small lists.
Where to go from here
That's genuinely all the linear algebra the next lessons assume. For the visual, animated version of these ideas, the gold standard is 3Blue1Brown's Essence of Linear Algebra series (on the Sources page). Next on the path: qubits and superposition.
The formal track
State vectors, normalization and bases are developed properly in the intermediate Academy module Mathematical Language of Qubits; Hilbert spaces, operators and composite systems in the advanced module Quantum Information Formalism.