Linear Algebra and Its Applications · Problem 1.4.3

An inner product pairs a row vector with a column vector to produce a single number—multiply matching entries and sum them up. The first example shows how $\mathbf{v} \cdot \mathbf{v}$ gives $\|\mathbf{v}\|^2$, a scalar measuring the vector's magnitude squared. The third product flips the order: column times row creates an entire matrix, where each entry is one component times another. Think of it as "outer product"—it builds a 2D table rather than collapsing to a number.

An inner product pairs a row vector with a column vector to produce a single number—multiply matching entries and sum them up. The first example shows how $\mathbf{v} \cdot \mathbf{v}$ gives $\|\mathbf{v}\|^2$, a scalar measuring the vector's magnitude squared. The third product flips the order: column times row creates an entire matrix, where each entry is one component times another. Think of it as "outer product"—it builds a 2D table rather than collapsing to a number.