List comprehensions are a concise way to create lists in Python. They allow you to generate a new list by performing some operation on each element of an existing iterable (such as a list or a string) or by applying a conditional statement. List comprehensions follow a specific syntax, which I’ll explain with examples.

Let’s start with a simple example. Suppose we have a list of numbers and we want to create a new list that contains the squares of each number. We can use a list comprehension to achieve this:

```
numbers = [1, 2, 3, 4, 5]
squares = [x**2 for x in numbers]
print(squares) # Output: [1, 4, 9, 16, 25]
```

In this example, we define the list comprehension `[x**2 for x in numbers]`. The expression `x**2` is the operation we want to perform on each element `x` in the `numbers` list. The resulting squares are collected and form a new list, which is assigned to the variable `squares`.

List comprehensions can also include conditional statements. Let’s modify our previous example to only include the squares of the odd numbers:

```
numbers = [1, 2, 3, 4, 5]
squares_of_odds = [x**2 for x in numbers if x % 2 != 0]
print(squares_of_odds) # Output: [1, 9, 25]
```

In this updated version, we added the conditional statement `if x % 2 != 0` at the end of the list comprehension. This statement filters out the even numbers from the original `numbers` list. Only the odd numbers are squared and added to the `squares_of_odds` list.

List comprehensions can also be nested, allowing you to create more complex lists. Here’s an example that generates a 2D matrix:

```
matrix = [[x for x in range(1, 4)] for y in range(3)]
print(matrix) # Output: [[1, 2, 3], [1, 2, 3], [1, 2, 3]
```

In this case, we have a nested list comprehension. The inner comprehension `[x for x in range(1, 4)]` generates a list `[1, 2, 3]`. The outer comprehension `[inner_comprehension for y in range(3)]` creates three instances of the inner list, resulting in a 2D matrix.

**Examples of list comprehensions, ordered from simplest to more complex, along with their explanations:**

**Example 1:** Creating a list of squared numbers from 1 to 5.

```
squares = [x**2 for x in range(1, 6)]
print(squares) # Output: [1, 4, 9, 16, 25]
```

**Explanation:** The list comprehension `[x**2 for x in range(1, 6)]` generates a new list by squaring each number `x` in the range from 1 to 5.

**Example 2: **Extracting vowels from a string.

```
word = "Hello, World!"
vowels = [char for char in word if char.lower() in 'aeiou']
print(vowels) # Output: ['e', 'o', 'o']
```

**Explanation:** The list comprehension `[char for char in word if char.lower() in ‘aeiou’]` loops over each character `char` in the string `word`. Only the characters that are vowels (both lowercase and uppercase) are added to the `vowels` list.

**Example 3:** Combining elements from two lists.

```
numbers = [1, 2, 3]
letters = ['A', 'B', 'C']
combined = [(num, letter) for num in numbers for letter in letters]
print(combined) # Output: [(1, 'A'), (1, 'B'), (1, 'C'), (2, 'A'), (2, 'B'), (2, 'C'), (3, 'A'), (3, 'B'), (3, 'C')]
```

**Explanation:** This example shows a nested list comprehension. The outer comprehension `[… for num in numbers for letter in letters]` iterates over each number in `numbers` and for each number, the inner comprehension `[num, letter] for letter in letters` creates a tuple of the number and each letter in `letters`. The result is a list of all possible combinations of numbers and letters.

**Example ****4****:** Converting a list of integers to strings.

```
numbers = [1, 2, 3, 4, 5]
number_strings =
number_strings = [str(num) for num in numbers]
print(number_strings) # Output: ['1', '2', '3', '4', '5']
```

**Explanation:** The list comprehension `[str(num) for num in numbers]` converts each integer `num` in the `numbers` list to a string using the `str()` function.

**Example ****5****: **Creating a list of even numbers using a conditional statement.

```
numbers = [1, 2, 3, 4, 5]
even_numbers = [num for num in numbers if num % 2 == 0]
print(even_numbers) # Output: [2, 4]
```

**Explanation: **The list comprehension `[num for num in numbers if num % 2 == 0]` filters the numbers from the `numbers` list and only adds the even numbers to the `even_numbers` list.

**Example ****6****:** Flattening a 2D matrix into a 1D list.

```
matrix = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]
flattened = [num for row in matrix for num in row]
print(flattened) # Output: [1, 2, 3, 4, 5, 6, 7, 8, 9]
```

**Explanation: **This example demonstrates a nested list comprehension. The outer comprehension `[num for row in matrix for num in row]` iterates over each row in the `matrix` list, and for each row, the inner comprehension `[num for num in row]` retrieves each element `num` from that row. The result is a flattened list containing all the elements from the 2D matrix.

**Example ****7:**Generating a list of prime numbers in a given range using a helper function.

```
def is_prime(n):
if n < 2:
return False
for i in range(2, int(n**0.5) + 1):
if n % i == 0:
return False
return True
# Solution with list comprehension
start = 1
end = 20
primes = [num for num in range(start, end+1) if is_prime(num)]
print(primes) # Output: [2, 3, 5, 7, 11, 13, 17, 19]
```

**Explanation:** Similar to a previous example, this list comprehension `[num for num in range(start, end+1) if is_prime(num)]` generates a list of prime numbers within a given range. The `is_prime()` function is used to determine if each number is prime.

**Example ****8****:** Filtering prime numbers from a range.

```
def is_prime(n):
if n < 2:
return False
for i in range(2, int(n**0.5) + 1):
if n % i == 0:
return False
return True
# Solution with list comprehension
numbers = range(1, 20)
primes = [num for num in numbers if is_prime(num)]
print(primes) # Output: [2, 3, 5, 7, 11, 13, 17, 19]
```

**Explanation: **In this example, we define a helper function `is_prime(n)` to check if a number `n` is prime. The list comprehension `[num for num in numbers if is_prime(num)]` generates a list of numbers from the `numbers` range for which the `is_prime()` function returns `True`.