Math for Data Science & ML

Start with the native identity matrix preset. Its two basis vectors span the plane, and a three by three illustrative grid remains a square grid. Now select the native singular preset. The two columns coincide. Every displayed input is multiplied by the same source matrix, so all nine transformed grid points lie on one diagonal line. The determinant falls from one to zero. This is a causal comparison of two exact held states, using sixty pixels per coordinate unit. The source random scatter is preserved on the interactive page; these displayed grid inputs are explicit illustrative coordinates. A zero determinant tells you that this transformation loses one dimension and collapses signed area. Switch to calculus, choose the paraboloid, and start at the native reset position, one point eight and one point five. The loss is zero point eight x squared plus one point five y squared. Its gradient there is two point eight eight and four point five. Compare the first Step action at learning rates zero point zero eight and zero point one six. Previous velocity is zero, so the momentum term contributes zero to this first update. The resulting positions are one point five six nine six, one point one four, and one point three three nine two, zero point seven eight. The true new loss is evaluated at each new position, using the displayed formula. Both bars use seventy pixels per loss unit. The source step history attaches the previous position loss to the new position; this explainer explicitly computes the true new loss rather than endorsing that lagged metric. A larger first step lowers this particular quadratic loss more, without claiming that larger learning rates always help. Reset the Bayesian panel to its actual Beta three, three prior. Before observing a coin, its mean is one half. Press the native Heads button once. The source adds one to alpha, producing Beta four, three. Its posterior mean becomes four sevenths, approximately zero point five seven one four two nine. The two held density curves are independently evaluated Beta probability densities, with the same horizontal and vertical scales. Their total probability integrates to one, although the plotted sampled polyline is an approximation. The interior maximum a posteriori value changes from one half to three fifths. This MAP formula is valid here because both parameters exceed one. The source uses a mean fallback at boundary parameter cases; that fallback is not being claimed as the correct boundary MAP. These are hypothetical native coin observations, not empirical evidence about an external coin.