What is the Row Vector Rule for Computing Ax?

The Row Vector Rule for Computing Ax is a fundamental principle in linear algebra that describes how to calculate each entry of the vector resulting from the multiplication of a matrix A by a vector x.

Understanding the Row Vector Rule

When you multiply a matrix A by a column vector x, the result is another column vector, denoted as Ax. The size of the resulting vector depends on the size of the matrix A. Specifically, if A is an m x n matrix (m rows, n columns) and x is an n x 1 vector (n entries), then the product Ax is defined, and the result will be an m x 1 vector (m entries).

The Row Vector Rule provides a direct method for finding each individual entry of this resulting vector Ax.

According to the rule:
If the product Ax is defined, then the i th entry in Ax is the sum of the products of corresponding entries from row i of A and from the vector x.

Let's break this down:

• "If the product Ax is defined": This is the prerequisite. As mentioned, Ax is defined only if the number of columns in matrix A equals the number of entries (rows) in vector x.
• "the i th entry in Ax": This refers to the entry in the i-th position (from the top) of the resulting vector Ax. The rule tells you how to calculate the first entry (i=1), the second entry (i=2), and so on, up to the m-th entry.
• "is the sum of the products of corresponding entries": This describes the calculation itself. For the i-th entry of Ax, you take the i-th row of matrix A and pair its entries with the entries of vector x.
• "from row i of A and from the vector x": You multiply the first entry of row i in A by the first entry of vector x, the second entry of row i by the second entry of vector x, and continue this for all entries. Finally, you add all these individual products together. This sum gives you the value of the i-th entry in the resulting vector Ax.

This process is equivalent to computing the dot product of the i-th row of matrix A (viewed as a vector) and the vector x.

Illustrating the Rule with an Example

Let's consider a simple example to see the Row Vector Rule in action.

Suppose matrix A is:

A = | 1  5  2 |
    | 4 -1  3 |

This is a 2x3 matrix (2 rows, 3 columns).

And vector x is:

x = | 2 |
    | 0 |
    |-3 |

This is a 3x1 vector (3 entries).

Since the number of columns in A (3) equals the number of entries in x (3), the product Ax is defined. The result will be a 2x1 vector.

Let's use the Row Vector Rule to find the entries of Ax:

The resulting vector Ax will have 2 entries.

1. Calculating the 1st entry of Ax:

   • Take the 1st row of A: [1 5 2]
   • Take vector x: [2 0 -3]
   • Multiply corresponding entries and sum them:
     (1 * 2) + (5 * 0) + (2 * -3)
     = 2 + 0 - 6
     = -4
   • So, the 1st entry of Ax is -4.

2. Calculating the 2nd entry of Ax:

   • Take the 2nd row of A: [4 -1 3]
   • Take vector x: [2 0 -3]
   • Multiply corresponding entries and sum them:
     (4 * 2) + (-1 * 0) + (3 * -3)
     = 8 + 0 - 9
     = -1
   • So, the 2nd entry of Ax is -1.

Therefore, the resulting vector Ax is:

Ax = | -4 |
     | -1 |

Why is this Rule Useful?

• Clear Calculation: It provides a step-by-step method for computing each specific entry in the result vector.
• Connects Concepts: It highlights the relationship between matrix-vector multiplication and the dot product of vectors. Each entry in Ax is the dot product of a row of A and the vector x.
• Foundation: Understanding this rule is crucial for grasping more complex matrix operations, such as matrix-matrix multiplication, which can be seen as applying the row vector rule for each column of the second matrix.

The Row Vector Rule is a fundamental concept for performing matrix-vector multiplication and understanding the underlying operations in linear algebra.