You will be able to
- Unroll a fixed linear state-space system.
- Explain why input-dependent transitions change the fixed-kernel interpretation.
State and readout
A discrete system hₜ=Ahₜ₋₁+Bxₜ, yₜ=Chₜ maps input through state and then a readout. A is a transition matrix, B injects input and C selects output. With h₋₁=0 and fixed matrices, unrolling gives a convolution-like sum with lag kernel CAᵏB. Nonzero initial state adds CAᵗ⁺¹h₋₁.
Modern models are more than fixed decay
Structured state-space models exploit constraints on transitions and kernels for long sequences. Mamba adds input-dependent selection and a hardware-aware scan; that is not generally a single fixed LTI convolution. The lab visualises a fixed two-dimensional rotating decay with impulse injection and no trained readout.
Worked example
For scalar A=0.5,B=C=1 and x=[1,2], h₀=1 and h₁=2.5. The convolution interpretation gives y₁=0.5×1+1×2=2.5.
A common trap
State-space models and attention are not interchangeable formulas. Their state, retrieval and input-dependence assumptions differ.
Predict, then experiment
What distinguishes a coordinate oscillation from forgetting? Before touching the controls, say or write your prediction.
Interactive experiment
An impulse leaves a rotating memory
How do decay and rotation control the response to one input?
State starts at [1,0], then evolves as hₜ₊₁=ρR(θ)hₜ with no further input. Solid blue and dashed amber are the two state coordinates, not the output of a trained model.
Your experiment
Hold decay 0.95 and compare rotation 0 and 0.8 rad/step. Observe the same norm decay despite different coordinate patterns. Explain what a learned C would change.
Change one parameter at a time. Compare what you observed with your prediction and record an explanation in your notes.
Check your understanding
Try from memory first. Hints and feedback are part of learning, not a penalty. These checks are for self-study, not a graded qualification.
Reference answers (after your attempt)
- the fixed-LTI-kernel interpretation no longer generally applies — Correct: selection changes the transition sequence.
- 2.5 — h₀=1; h₁=0.5×1+2=2.5.
Apply it: Compare representations of the same system
Use paper, your notes, or a local notebook/editor. Code examples are Python 3 starting points to run yourself; this page does not execute code or automatically assess the practical task.
- Compute the scalar recurrence for x=[1,2,0].
- Compute the lag-kernel sum at each step.
- Explain what changes if A depends on xₜ.
Reference solution & review criteria
Both fixed-system calculations give [1,2.5,1.25]. If A varies with the input, different transition products apply at different times; the one fixed convolution kernel no longer generally represents the operator.
Your notebook
Notes and progress stay in this browser. They are not sent to the server; ordinary web-server request logs still operate. Export a backup from the course page.
Completion requires both correct checks and your practical-task acknowledgement.
Go deeper
Sources support further study; the worked examples and questions here are original. Browser experiments are small, disclosed models—not evidence of production-model quality or hardware performance.