Question

Write and test a function innerProd \((x, y)\) that computes the inner product of two (same length) lists. The inner product of \(x\) and \(y\) is computed as: \\[\sum_{i=0}^{n-1} x_{i} y_{i}\\]

Short answer

Define `innerProd(x, y)`, initialize total, multiply and sum corresponding elements, return `total`.

Step-by-step solution

Define the Function Signature

We will start by defining the function `innerProd` which takes two parameters, `x` and `y`, both expected to be lists of the same length. This definition sets up the function for processing the lists to compute the inner product.

Initialize the Accumulator

Inside the function, initialize a variable `total` to zero. This variable will be used to accumulate the sum of products of corresponding elements from the lists `x` and `y`.

Iterate Over Elements of the Lists

Use a `for` loop to iterate over the indices of the lists. Within the loop, multiply the elements `x[i]` and `y[i]` for each index `i` and add the result to `total`. This loop ensures that each pair of corresponding elements from the two lists is multiplied and added to the `total` variable.

Return the Result

After completing the loop, return the value of `total` from the function. This value is the computed inner product of the two lists.

Test the Function

Test the `innerProd` function with example lists to ensure it works correctly. For instance, calling `innerProd([1, 2, 3], [4, 5, 6])` should return `32`, because the inner product is \((1 \times 4) + (2 \times 5) + (3 \times 6) = 32\).

Key concepts

Function Definition in Python

In Python programming, a function is a reusable block of code that performs a specific task. When you define a function, you create it with a signature, essentially a name, followed by parameters in parentheses. This allows for specific inputs. To define a function, use the `def` keyword, followed by the function name and the parameters it takes. For example:
`def innerProd(x, y):`
Here, `innerProd` is the function name, while `x` and `y` are parameters representing the lists for which we will calculate the inner product. Ensuring clear and precise naming in the function definition helps users understand its purpose.

Understanding For Loops

A `for` loop in Python is used for iterating over a sequence, such as a list or a range of numbers, allowing you to execute a set of statements for each item in the sequence. In the context of the inner product computation, the loop facilitates the traversal through each element of the input lists.

• To use a `for` loop, you can iterate over the range of indices that match the length of the lists.
• For example, `for i in range(len(x)):` gives access to each corresponding index of x and y.

Within this loop, you perform operations on elements at the same position across both lists, making `for` loops a fundamental tool for list processing and manipulation in many numerical computations.

Simplifying List Manipulation

List manipulation involves modifying lists using various methods and operations. In Python, lists are versatile and can store ordered sequences of items. They allow for operations such as accessing, modifying, and iterating over elements. In our exercise, list manipulation involves accessing list elements by their indices to compute their products.

• Access elements using their index, for example, `x[i]` retrieves the item at position `i` in list `x`.
• List comprehensions provide a succinct way to apply operations across list elements.

Understanding list manipulation mechanisms is key when performing complex tasks like the computation of inner products.

Inner Product Computation Explained

An inner product, also known as a dot product, is a mathematical operation that multiplies corresponding elements of two equal-length sequences and sums the results. The formula for computing the inner product of two lists, `x` and `y`, is given by:
\[ \sum_{i=0}^{n-1} x_{i} y_{i} \]In Python, this is achieved by iterating over the elements of both lists using equal indices, thereby multiplying the pair at each index and accumulating the sum.

• Multiply corresponding elements, such as `x[i] * y[i]`.
• Accumulate the results in a variable, often initialized as zero.

By the end of the iteration, this cumulative sum is returned, signifying the inner product of the two lists. This operation is a cornerstone in fields requiring numerical analysis and linear algebra.