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The span of two arbitrary vectors $\vec{v}$ and $\vec{w}$ that do not lie on the same line is the entire 2D plane. This is achieved by taking all possible linear combinations $a\vec{v} + b\vec{w}$ where scalars $a$ and $b$ vary continuously across all real numbers.

Conditions: Two vectors $\vec{v}$ and $\vec{w}$ are nonzero; Vectors $\vec{v}$ and $\vec{w}$ are non-collinear (point in different directions)

The mean value theorem states that if a function $f$ is continuous on the closed interval $[a,b]$ and differentiable on the open interval $(a,b)$, then there exists at least one point $c$ in $(a,b)$ such that the instantaneous rate of change (derivative) at $c$ equals the average rate of change over the interval. Mathematically, this is expressed as $f'(c) = \frac{f(b)-f(a)}{b-a}$.

Conditions: $a < b$; $f$ is continuous on $[a,b]$; $f$ is differentiable on $(a,b)$

An undirected connected graph with n vertices has at least n-1 edges. This minimum is achieved when the graph forms a tree structure.

Conditions: The graph is undirected.; The graph is connected.; The graph has n vertices.

Geometrically, $\epsilon$ defines a horizontal band around the limit value $L$ on the y-axis, representing the target range for function outputs. $\delta$ defines a vertical interval around the input value $a$ on the x-axis, representing the allowable range for inputs.

Conditions: The graph is plotted on a Cartesian coordinate system.; $\epsilon$ is the half-height of the band around $L$.; $\delta$ is the half-width of the interval around $a$.

Import `MinMaxScaler` from `sklearn.preprocessing`, instantiate it, and call `fit_transform` on the selected DataFrame columns. This fits the scaler to the data (learning min/max) and transforms the values to the [0, 1] range.

Conditions: Input data is numerical.; Scikit-learn library is installed.

The intuitive concept says $f(x)$ 'gets closer' to $L$ as $x$ 'gets closer' to $a$. The epsilon-delta definition formalizes this by replacing vague 'closeness' with precise quantitative tolerances: $\epsilon$ measures output closeness to $L$, and $\delta$ measures input closeness to $a$.

Conditions: The intuitive notion of 'approaching' is qualitative.; The formal definition introduces universal and existential quantifiers over positive real numbers.

Once convergence to a limit $L$ is established, pass to the limit in the recurrence relation $a_{n+1}=\frac{1}{3}(a_n+1)$. Since both $a_n$ and $a_{n+1}$ approach $L$, substitute $L$ for both terms to get $L=\frac{1}{3}(L+1)$, and then solve this linear equation for $L$.

Conditions: The sequence is known to converge to some real number $L$.; The recurrence relation is $a_{n+1}=\frac{1}{3}(a_n+1)$.

When integrating a multivariable function with respect to one variable, the other variables are treated as constants. Therefore, the 'constant' of integration can be any function that depends only on the other variables, which in this case is y.

Conditions: The integration is with respect to x.; The function M depends on both x and y.

The PCA algorithm subtracts the mean from the data to centralize it, meaning that the mean of all points should be zero. This step prepares the data for the variance-based step that follows, making the formula for the scatter matrix simpler.

Conditions: The data must first be represented as a matrix X.; The subtraction is by the mean $\overline{X}$.; After this step, the video uses $\overline{X} = 0$ in the scatter-matrix discussion.

The fundamental theorem relates accumulation and rate of change by stating that if $F'(x) = f(x)$, then the accumulated area $A(x)$ from a fixed point $a$ to $x$ is given by $A(x) = F(x) - F(a)$. It shows that differentiation and integration are inverse processes: the derivative of the accumulation function $A(x)$ recovers the integrand $f(x)$, and antiderivatives allow the evaluation of definite integrals.

Conditions: The function $f$ is continuous.; $F$ is an antiderivative of $f$ (i.e., $F' = f$).

This method approximates the volume under $f(x,y)=9-x^2-y^2$ by partitioning the square base $[-2,2] \times [-2,2]$ into four equal subsquares, each with area $\Delta A_k = 4$. Sample points such as the bottom-left corners $(-2,-2), (0,-2), (-2,0), (0,0)$ are selected, their function values summed, and the total multiplied by 4 to yield an approximate volume of 80. This is a discrete Riemann sum approximation, not the exact volume.

Conditions: The base region is $[-2,2] \times [-2,2]$.; The partition consists of four equal subsquares.; Sample points are chosen according to a specific rule (e.g., bottom-left corners).

Relabeling the blocks by inequivalent irreducible representations groups equivalent irreducible summands under a single label. This makes it possible to introduce a multiplicity number $a_j$ that records how many times the irreducible representation $\Gamma^j$ occurs in $\Gamma$, leading to the compact expression $\Gamma = \bigoplus_j a_j \Gamma^j$.

Conditions: The original block diagonal decomposition is available.; Equivalent irreducible summands are grouped under the same label.