Chapter 8: Introduction to NumPy

Sub-Chapter: NumPy Arrays and Operations — The Backbone of Numerical Computing

NumPy arrays (or ndarrays) are the core data structure in NumPy and the foundation of high-performance numerical computing in Python.
They provide efficient, vectorized operations on homogeneous data — enabling mathematical computations that are fast, expressive, and memory-efficient.

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🧩 1. What Makes NumPy Arrays Special?

Unlike Python lists, which store objects of mixed types and have high overhead, NumPy arrays:

• Store homogeneous data in contiguous memory.
• Support vectorized operations — no explicit loops.
• Enable broadcasting between different shapes.
• Are implemented in C for performance.

Comparison Example

import numpy as np
import time

nums = list(range(1_000_000))
arr = np.arange(1_000_000)

# Python list
start = time.time()
nums_result = [x * 2 for x in nums]
print("List:", time.time() - start)

# NumPy array
start = time.time()
arr_result = arr * 2
print("NumPy:", time.time() - start)

🚀 NumPy operations can be 50–100× faster than native Python loops.

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⚙️ 2. Creating Arrays

You can create NumPy arrays from Python sequences or using NumPy’s built-in constructors.

import numpy as np

a = np.array([1, 2, 3])                    # From list
b = np.array([[1, 2, 3], [4, 5, 6]])       # 2D array
zeros = np.zeros((2, 3))                   # All zeros
ones = np.ones((3, 3))                     # All ones
range_array = np.arange(0, 10, 2)          # Evenly spaced values
linspace = np.linspace(0, 1, 5)            # 5 values between 0 and 1
identity = np.eye(3)                       # 3×3 identity matrix
randoms = np.random.randint(0, 10, (2, 3)) # Random integers 0–9

Quick Reference: Array Creation

Function	Description	Example
np.array()	Convert list to ndarray	np.array([1,2,3])
np.zeros()	Create zeros	np.zeros((3,3))
np.ones()	Create ones	np.ones((2,4))
np.arange()	Range with step	np.arange(0,10,2)
np.linspace()	Evenly spaced	np.linspace(0,1,5)
np.eye()	Identity matrix	np.eye(3)
np.empty()	Uninitialized memory	np.empty((2,3))
np.random.rand()	Uniform random	np.random.rand(2,2)

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🔍 3. Array Properties and Attributes

Each NumPy array exposes useful metadata.

arr = np.array([[1, 2, 3], [4, 5, 6]])

print(arr.ndim)     # 2 dimensions
print(arr.shape)    # (2, 3)
print(arr.size)     # 6 elements
print(arr.dtype)    # int64
print(arr.itemsize) # Bytes per element
print(arr.nbytes)   # Total memory usage

🧠 All NumPy arrays are typed — mixing floats and ints will upcast automatically (e.g. int → float).

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🔢 4. Array Operations (Vectorization)

NumPy automatically applies operations element-wise — known as vectorization.

a = np.array([1, 2, 3])
b = np.array([4, 5, 6])

print(a + b)   # [5 7 9]
print(a * b)   # [4 10 18]
print(a ** 2)  # [1 4 9]
print(a / 2)   # [0.5 1. 1.5]

Comparison and Logical Operations

arr = np.array([10, 20, 30, 40])
print(arr > 25)         # [False False  True  True]
print(np.logical_and(arr > 10, arr < 40))

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🧮 5. Matrix and Linear Algebra Operations

NumPy supports advanced linear algebra directly.

A = np.array([[1, 2], [3, 4]])
B = np.array([[5, 6], [7, 8]])

print(A @ B)           # Matrix multiplication
print(np.dot(A, B))    # Same as A @ B
print(np.transpose(A)) # Transpose
print(np.linalg.inv(A))# Inverse
print(np.linalg.det(A))# Determinant

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🧠 6. Statistical and Aggregate Functions

arr = np.array([[1, 2, 3], [4, 5, 6]])

print(arr.sum())         # 21
print(arr.mean())        # 3.5
print(arr.std())         # Standard deviation
print(arr.min(axis=0))   # Column-wise min
print(arr.max(axis=1))   # Row-wise max

Use axis=0 for columns and axis=1 for rows.

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🔄 7. Reshaping and Transposing

You can change the shape of an array without copying its data.

matrix = np.arange(12).reshape(3, 4)

print(matrix.shape)      # (3, 4)
print(matrix.reshape(4, 3))
print(matrix.flatten())  # 1D copy
print(matrix.ravel())    # 1D view (no copy)
print(matrix.T)          # Transposed

Shape Diagram

Original (3×4)
[[0, 1, 2, 3],
 [4, 5, 6, 7],
 [8, 9, 10, 11]]

Reshaped (4×3)
[[0, 1, 2],
 [3, 4, 5],
 [6, 7, 8],
 [9,10,11]]

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📈 8. Broadcasting Explained

Broadcasting allows operations between arrays of different shapes by expanding smaller arrays automatically.

matrix = np.array([[1, 2, 3], [4, 5, 6]])
vector = np.array([10, 20, 30])

print(matrix + vector)

Output:

[[11 22 33]
 [14 25 36]]

Broadcasting Rules

1. Compare dimensions from right to left.
2. Dimensions are compatible if they are equal or one of them is 1.
3. If incompatible, a ValueError occurs.

Visual Explanation

Matrix (2×3): [[1 2 3]
               [4 5 6]]
Vector (1×3): [10 20 30]
→
[[11 22 33]
 [14 25 36]]

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🧩 9. Combining and Splitting Arrays

a = np.array([[1, 2], [3, 4]])
b = np.array([[5, 6]])

combined = np.vstack((a, b))  # Vertical stack
hstacked = np.hstack((a, b.T))# Horizontal stack
split = np.split(a, 2)        # Split into two arrays

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⚡ 10. Real-World Example — Data Normalization

data = np.array([[10, 20, 30],
                 [40, 50, 60],
                 [70, 80, 90]])

mean = np.mean(data, axis=0)
std = np.std(data, axis=0)
normalized = (data - mean) / std

print(normalized)

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🧾 11. Quick Reference Tables

Array Attributes

Attribute	Description	Example Output
ndim	Number of dimensions	2
shape	Tuple of array size	(3, 4)
size	Total number of elements	12
dtype	Data type	float64
itemsize	Bytes per element	8

Common Functions

Category	Function	Purpose
Creation	np.zeros(), np.ones(), np.arange(), np.random.rand()	Create arrays
Math	np.add, np.multiply, np.power	Element-wise math
Statistics	np.mean, np.std, np.min, np.max	Aggregation
Shape	reshape, flatten, ravel, resize	Manipulation
Linear Algebra	np.dot, np.linalg.inv, np.linalg.det	Matrix operations

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🧭 12. Best Practices

✅ Use vectorized operations instead of loops.
✅ Prefer broadcasting to manual repetition.
✅ Always check dtype when mixing types.
✅ Avoid unnecessary copies with .ravel() instead of .flatten().
✅ Use axis wisely for column/row aggregation.
✅ Use np.random for reproducible test data.

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🧠 Summary

Concept	Description	Example
Array Creation	From lists or constructors	np.array([[1,2],[3,4]])
Element-wise Ops	Fast, vectorized math	a + b, a * 2
Matrix Ops	Dot, transpose, inverse	A @ B, A.T
Reshape	Change structure	arr.reshape(3,4)
Broadcasting	Align shapes	matrix + vector

NumPy arrays turn Python into a high-performance, data-parallel computation environment.
Mastering them unlocks everything from basic math to advanced machine learning pipelines.