Problem 50 - Discrete Mathematics with Applications

Question

Expand each. $$\sum_{j=1}^{2} \sum_{i=1}^{3} a_{i j}$$

Step-by-Step Solution

Answer

The short answer is: Expanding the given expression, we have: \[ \sum_{j=1}^{2} \sum_{i=1}^{3} a_{i j} = (a_{11} + a_{21} + a_{31}) + (a_{12} + a_{22} + a_{32}) \]

Step 1: Expand the outer summation

Expand the outer summation, which goes from j=1 to 2. For each value of j, we will have an inner sum: $$ \sum_{j=1}^{2} \sum_{i=1}^{3} a_{i j} = \left(\sum_{i=1}^{3} a_{i1}\right) + \left(\sum_{i=1}^{3} a_{i2}\right) $$

Step 2: Expand the inner summations

Next, expand the inner summations for each value of i from 1 to 3: $$ \left(\sum_{i=1}^{3} a_{i1}\right) = a_{11} + a_{21} + a_{31} $$ and $$ \left(\sum_{i=1}^{3} a_{i2}\right) = a_{12} + a_{22} + a_{32} $$

Step 3: Combine the expansions

Finally, substitute the expanded expressions back into the original expression: $$ \sum_{j=1}^{2} \sum_{i=1}^{3} a_{i j} = \left(a_{11} + a_{21} + a_{31}\right) + \left(a_{12} + a_{22} + a_{32}\right) $$

Key Concepts

Outer Summation

The concept of an outer summation might initially appear daunting, especially when paired with inner summations. But don't worry, we're here to simplify it. The outer summation is the broader operation that encompasses an inner summation. Think of it as the larger loop that wraps around another, smaller loop within it. Let's see how this works in practice.

In our example, the outer summation operates over the index \( j \), ranging from 1 to 2. This means that we repeat a series of operations, once for each value of \( j \) in this range:

• When \( j = 1 \), we compute the inner summation over \( i \).
• Then, \( j \) increments to 2, and we again compute the inner summation over \( i \).

This provides a clear structure for breaking down complex summations into manageable portions, iteratively summing elements in two dimensions.

Inner Summation

The inner summation is where we start to see the details unfold. It happens inside the outer summation and specifically iterates over a specific index, typically denoted by \( i \).

In our example, for each fixed value of \( j \), the inner summation calculates the sum of terms where \( i \) varies from 1 to 3:

• When \( j = 1 \), the inner summation becomes: \( a_{11} + a_{21} + a_{31} \).
• For \( j = 2 \), it progresses as: \( a_{12} + a_{22} + a_{32} \).

This inner operation is akin to focusing on a row in a matrix, summing up elements as it sweeps across specific column positions. The main goal of this step is to simplify the summation structure so that each little part can be added together effortlessly.

Matrix Expansion

Matrix expansion brings everything together. It refers to rewriting or unfolding the double summation into a more explicit form. This involves expanding both the inner and outer summations, as we did in the previous steps.

When performing matrix expansion, it's crucial to pay attention to every index involved, ensuring no term is left out. From our expansion, we end up with:
\[\sum_{j=1}^{2} \sum_{i=1}^{3} a_{ij} = \left(a_{11} + a_{21} + a_{31}\right) + \left(a_{12} + a_{22} + a_{32}\right)\]
This expression organizes the matrix elements into a comprehensive sum, providing a complete representation of the double summation process. It helps in visualizing and simplifying equations, forming a released form that is much easier to manipulate in further mathematical operations.