(a) Let A be an m ×n matrix. Let 𝐞j denote the 1 ×n column vector with a single 1 in the j^th e

step 1 So we know with matrix multiplication that the number of columns in our first matrix must be equ

(a) Let A be an m ×n matrix. Let 𝐞j denote the 1 ×n column...

(a) Let $A$ be an $m \times n$ matrix. Let $\mathbf{e}_j$ denote the $1 \times n$ column vector with a single 1 in the $j^{\text {th }}$ entry, as in (1.44). Explain why the product $A \mathbf{e}_j$ equals the $j^{\text {th }}$ column of $A$. (b) Similarly, let $\hat{\mathbf{e}}_i$ be the $1 \times m$ column vector with a single 1 in the $i^{\text {th }}$ entry. Explain why the triple product $\hat{\mathbf{e}}_i^T A \mathbf{e}_j=a_{i j}$ equals the $(i, j)$ entry of the matrix $A$.

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Key Concepts

Standard Basis Vectors

Standard basis vectors are vectors with one entry equal to one and all other entries equal to zero. They provide a natural way to 'select' specific components in vector spaces. When these vectors are used in operations such as matrix multiplication, they act like selectors that pick out entire rows or columns, depending on their position in the multiplication sequence.

Matrix Multiplication as a Linear Combination of Columns

Matrix multiplication can be viewed as forming linear combinations of the matrix's columns. Multiplying a matrix by a standard basis vector on the right yields the corresponding column of the matrix. This is because the basis vector has a one in a single entry and zeros elsewhere, effectively eliminating all other columns from contributing to the product.

Entry Extraction through Triple Product

When a transposed standard basis vector (which is used to select a particular row) is multiplied by the matrix and then by another standard basis vector (which selects a particular column), the result is a single scalar. This scalar is the entry in the matrix located at the intersection of the selected row and column, thus directly extracting the (i, j) entry of the matrix.

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