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Subtract $a_n$ from both sides of the recurrence relation. Distribute the $\frac{1}{3}$ and combine like terms to simplify the expression to $-\frac{2}{3}a_n + \frac{1}{3}$.

Conditions: The recurrence relation is $a_{n+1}=\frac{1}{3}(a_n+1)$.; The goal is to analyze the sign of the difference to determine monotonicity.

To calculate the total cost, you must look at the original cost matrix, not the reduced matrix. Identify the positions of the circled zeros in the reduced matrix, find the corresponding entries in the original matrix, and sum those values.

Conditions: The optimal assignment has been determined from the reduced matrix.; The original cost matrix is available.

A transition diagram represents the possible states of a system as nodes and the allowed transitions between states as directed arrows. The number attached to an arrow gives the probability of moving from the tail state to the head state in one step.

Conditions: The diagram is being used for a Markov chain.; Each arrow label is interpreted as a one-step transition probability.

To find the point on the curve at $x = -1$, substitute $x = -1$ into the curve's equation $y = x^2 - 4$. This gives $y = (-1)^2 - 4 = 1 - 4 = -3$.

Conditions: The curve is defined by the equation $y = x^2 - 4$.; The x-coordinate of the desired point is given as $-1$.; The point must lie on the curve.

Sensitivity is the probability of a positive test result given that the person has the disease ($P(+\mid D)$), while specificity is the probability of a negative test result given that the person does not have the disease ($P(-\mid D^c)$). In the model, both are assumed to be $95\%$, but they apply to different subgroups: the sick versus the healthy.

Conditions: The test has $95\%$ sensitivity.; The test has $95\%$ specificity.; The model distinguishes between diseased and non-diseased populations.

Gradient descent oscillates when features are not scaled because features with vastly different ranges cause the gradients with respect to their corresponding parameters to differ greatly. This leads to small step sizes for parameters of small-scale features and large step sizes for parameters of large-scale features, resulting in an inefficient, zigzagging path toward the minimum.

Conditions: Used in linear regression optimization with gradient descent.; Features have significantly different scales (e.g., one ranges from 0 to 1, another from 100 to 1000).

A set of vectors is linearly dependent if one of the vectors is redundant, meaning it provides no new directional freedom. Mathematically, this occurs when a vector lies perfectly flat on the plane (or line) already created by the others, allowing it to be written as a combination of the others.

Conditions: The set contains at least two vectors.; One vector lies in the span of the other vectors.

The determinant is 1. This is calculated using the formula for a 2x2 matrix where the columns are $\begin{pmatrix} 3 & 2 \\ 1 & 1 \end{pmatrix}$.

Conditions: The matrix has columns $(3,1)$ and $(2,1)$.; The determinant formula $ad-bc$ is applied correctly to the column entries.

Local linearity means that although a transformation is globally nonlinear and complex, it appears linear when viewed in a sufficiently small neighborhood around a specific point. The Jacobian matrix captures this local linear behavior.

Conditions: The transformation is nonlinear globally.; The analysis is restricted to a small neighborhood around a specific point.; The zoom level is high enough that curvature is negligible.

In Ax=b, rows of A contain the equation coefficients, x lists the unknowns, and b lists the constants. The dimensions must match.

Conditions: The system is linear.; Dimensions of the coefficient matrix, variable vector, and constant vector are compatible.

To finish on trial x, the first $x-1$ trials must contain exactly $r-1$ successes, and the x-th trial must be a success. The probability is calculated as the number of ways to arrange $r-1$ successes in $x-1$ trials, multiplied by the probability of that specific sequence.

Conditions: Trials are independent.; Success probability p is constant.; $x \ge r$.

$R_0=1$ is the threshold separating outbreak decline from growth. If $R_0 < 1$, each case infects fewer than one other person on average, causing transmission chains to break and the outbreak to fade.

Conditions: The video states the growth side only until immunity builds up.; The example value shown is $R_0=2.5$.