3Blue1Brown: Linear Combinations, Span, and Basis Vectors - Chapter 2 Quiz

This quiz assesses a deep conceptual and applied understanding of linear combinations, vector span, and basis vectors, as introduced in the 3Blue1Brown 'Essence of Linear Algebra' Chapter 2 video. Questions cover interpreting vector coordinates, defining and visualizing span in 2D and 3D, identifying linear dependence and independence, and applying the technical definition of a basis. Challenging questions require nuanced reasoning and an ability to connect abstract definitions to geometric intuition.

Key formulas/concepts involved:

• Linear Combination: A vector $\vec{v}$ is a linear combination of vectors $\vec{v_1}, \vec{v_2}, \dots, \vec{v_k}$ if it can be expressed as: $\vec{v} = c_1\vec{v_1} + c_2\vec{v_2} + \dots + c_k\vec{v_k}$ where $c_1, c_2, \dots, c_k$ are scalars.
• Span: The span of a set of vectors $S = \{\vec{v_1}, \vec{v_2}, \dots, \vec{v_k}\}$ is the set of all possible linear combinations of those vectors. It is denoted as $\text{span}(S)$.
• Linear Dependence: A set of vectors $S = \{\vec{v_1}, \vec{v_2}, \dots, \vec{v_k}\}$ is linearly dependent if at least one vector in $S$ can be expressed as a linear combination of the others, or equivalently, if there exist scalars $c_1, c_2, \dots, c_k$, not all zero, such that: $c_1\vec{v_1} + c_2\vec{v_2} + \dots + c_k\vec{v_k} = \vec{0}$
• Linear Independence: A set of vectors is linearly independent if it is not linearly dependent, meaning the only way to form the zero vector is if all scalars in the linear combination are zero.
• Basis: A basis for a vector space $V$ is a set of linearly independent vectors that span $V$. For a given vector space of dimension $n$, a basis will always consist of exactly $n$ linearly independent vectors.