Shape After Indexing¶

Indexing and slicing affect array dimensions differently.

Index vs Slice¶

Single index reduces dimensions; slice preserves them.

1. Single Index¶

```python import numpy as np

def main(): a = np.zeros((8, 8, 8, 8)) print(f"{a[1].shape = }")

if name == "main": main() ```

Output:

a[1].shape = (8, 8, 8)

2. Slice of One¶

```python import numpy as np

def main(): a = np.zeros((8, 8, 8, 8)) print(f"{a[1:2].shape = }")

if name == "main": main() ```

Output:

a[1:2].shape = (1, 8, 8, 8)

3. Key Difference¶

a[1] removes a dimension; a[1:2] keeps it with size 1.

Mixed Operations¶

Combining indices and slices on different axes.

1. Index Two Axes¶

```python import numpy as np

def main(): a = np.zeros((8, 8, 8, 8)) print(f"{a[1, :, 3, :].shape = }")

if name == "main": main() ```

Output:

a[1, :, 3, :].shape = (8, 8)

2. Index and Slice¶

```python import numpy as np

def main(): a = np.zeros((8, 8, 8, 8)) print(f"{a[1:2, :, 3, :].shape = }")

if name == "main": main() ```

Output:

a[1:2, :, 3, :].shape = (1, 8, 8)

3. All Slices¶

```python import numpy as np

def main(): a = np.zeros((8, 8, 8, 8)) print(f"{a[1:2, :, 3:4, :].shape = }")

if name == "main": main() ```

Output:

a[1:2, :, 3:4, :].shape = (1, 8, 1, 8)

Complete Comparison¶

Side-by-side comparison of all cases.

1. Summary Table¶

```python import numpy as np

def main(): a = np.zeros((8, 8, 8, 8)) print(f"{a[1].shape = }") print(f"{a[1:2].shape = }") print(f"{a[1, :, 3, :].shape = }") print(f"{a[1:2, :, 3, :].shape = }") print(f"{a[1:2, :, 3:4, :].shape = }")

if name == "main": main() ```

Output:

a[1].shape = (8, 8, 8) a[1:2].shape = (1, 8, 8, 8) a[1, :, 3, :].shape = (8, 8) a[1:2, :, 3, :].shape = (1, 8, 8) a[1:2, :, 3:4, :].shape = (1, 8, 1, 8)

2. Dimension Rule¶

Each integer index removes one dimension; each slice keeps it.

Practical Impact¶

Understanding shape changes is crucial for array operations.

1. Broadcasting¶

Shape mismatches cause broadcasting errors; use slices to preserve dimensions.

2. Neural Networks¶

Batch dimensions must be preserved; use [0:1] instead of [0].

3. Matrix Operations¶

Some operations require 2D arrays; slicing maintains dimensionality.

Exercises¶

Exercise 1. Given a = np.zeros((6, 5, 4, 3)), predict the shape of each expression without running the code, then verify:

• a[0]
• a[0:1]
• a[2, :, 1]
• a[2:3, :, 1:2]

import numpy as np

a = np.zeros((6, 5, 4, 3))
print(a[0].shape)          # (5, 4, 3)
print(a[0:1].shape)        # (1, 5, 4, 3)
print(a[2, :, 1].shape)    # (5, 3)
print(a[2:3, :, 1:2].shape)# (1, 5, 1, 3)

Exercise 2. Create a 3D array a = np.zeros((4, 3, 2)). Extract a single row from the middle axis using an integer index (a[:, 1, :]) and using a slice (a[:, 1:2, :]). Print both shapes and explain why one has 2 dimensions and the other has 3.

import numpy as np

a = np.zeros((4, 3, 2))
with_index = a[:, 1, :]     # integer index removes axis 1
with_slice = a[:, 1:2, :]   # slice preserves axis 1

print(f"Integer index shape: {with_index.shape}")  # (4, 2)
print(f"Slice shape: {with_slice.shape}")          # (4, 1, 2)
# The integer index removes the dimension entirely,
# while the slice keeps it with size 1.

Exercise 3. Given images = np.random.randn(32, 3, 64, 64) (a batch of 32 RGB images), extract the first image preserving all 4 dimensions (shape (1, 3, 64, 64)) using a slice. Then extract it with an integer index and verify the shape is (3, 64, 64). Show how to restore the batch dimension using np.expand_dims.

import numpy as np

images = np.random.randn(32, 3, 64, 64)

first_slice = images[0:1]
print(f"With slice: {first_slice.shape}")  # (1, 3, 64, 64)

first_index = images[0]
print(f"With index: {first_index.shape}")  # (3, 64, 64)

restored = np.expand_dims(first_index, axis=0)
print(f"Restored: {restored.shape}")  # (1, 3, 64, 64)