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To find the coordinates of a vector in an alternative basis, express the vector as a linear combination of the new basis vectors. The scalars in this combination are the coordinates.

Conditions: The alternative basis vectors are linearly independent.; The target vector lies in the span of the alternative basis.

When using u-substitution in a definite integral, the original limits of integration in terms of $x$ must be converted into new limits in terms of $u$. This is done by substituting the original $x$-bounds into the substitution equation $u(x)$.

Conditions: The integral is a definite integral.; A substitution $u = u(x)$ is being used.; The original limits of integration are given in terms of $x$.

Eigenvectors are special because they do not rotate off their original line (span) during a transformation; they only stretch, shrink, or flip. The zero vector is excluded from this definition because it maps to itself under any linear transformation ($A\vec{0}=\vec{0}$), providing no directional information about the transformation's effect.

Conditions: Vector $\vec{v}$ must be non-zero ($\vec{v} \neq \vec{0}$); Transformation is linear

The vertical line test is a graphical method used to determine whether a curve represents a function of $x$. A graph represents a function if and only if no vertical line intersects the graph at more than one point.

Conditions: The graph is plotted on a standard Cartesian coordinate system.; The independent variable is $x$.

Valid linear transformations keep grid lines straight, parallel, and evenly spaced. Nonlinear distortions involve bending or curving of these lines.

Conditions: Observing the movement of a 2D coordinate grid.; Checking for straightness, parallelism, and equal spacing of lines.

The Euclidean algorithm finds the greatest common divisor (GCD) of two integers without needing to factor them. The process involves repeatedly performing long division: divide the larger number by the smaller number, then divide the previous divisor by the remainder, and continue this process.

Conditions: Applies to two integers.; Requires repeated long division.; Stops when the remainder is zero.

The substitution $u = 1 + \frac{9}{4}x$ is chosen because it is exactly the expression under the radical in the arc-length integral $\int_0^{32/9}\sqrt{1+\frac{9}{4}x}\,dx$. By setting $u$ to this inner expression, the complicated integrand simplifies to $\sqrt{u}$, which is much easier to integrate using the power rule.

Conditions: The integral to evaluate is $\int_0^{32/9}\sqrt{1+\frac{9}{4}x}\,dx$.; The integrand contains a composite expression under a square root.

No, a sample size of 30 is not strictly required. The video clarifies that $n \ge 30$ is merely a conservative rule of thumb.

Conditions: The discussion addresses the common $n \ge 30$ heuristic.; The video demonstrates CLT applicability with $n=20$.; The population distribution is not extremely skewed or heavy-tailed.

Substitute the given numerical values into the rearranged formula P(B|A) = $P(B)P$(A|B)/P(A). For example, if $P(B) = 1/21$, P(A|B) = $4/10$, and $P(A) = 24/210$, the calculation is ($1/21 * 4/10$) / ($24/210$).

Conditions: The formula used is the rearranged Bayes identity for P(B|A).; The given probabilities must be consistent with the formula's assumptions (e.g., $P(A) > 0$).

The video uses these three conditions because a generic quadratic polynomial has exactly three unknown coefficients. By requiring the approximation to match the function value, the first derivative (slope), and the second derivative (concavity) at the expansion point, the lecture creates a system of three equations that determines the three coefficients uniquely within the worked setup.

Conditions: The approximating polynomial is quadratic.; The matching point is $x=0$.; The target function and polynomial can be differentiated at least twice at that point.

The book and magnifying glass icons represent specific events in a probability space, serving as labels for the variables in Bayes' theorem. The formula shown is P(Book|Magnifier) = P(Book)P(Magnifier|Book)/P(Magnifier).

Conditions: The icons are used as event labels in the displayed formula.; No fixed letter-to-icon assignment is imposed across displays in the video.

Bayes' theorem is true because the joint probability of two events can be decomposed in two symmetric ways. The probability that both A and B occur is equal to the probability of A multiplied by the conditional probability of B given A, and also equal to the probability of B multiplied by the conditional probability of A given B.

Conditions: A and B are events in the same probability space.; Both A and B have positive probability for the ordinary conditionals used here.