Transpose
- Machine Learning course
- Beginner
- Lesson 34/112
- 5 min
In a matrix transpose, rows and columns are swapped.
Transpose
The transpose turns a matrix's rows into columns: (m, n) → (n, m). In NumPy it is written .T.
A = np.array([[1, 2, 3],
[4, 5, 6]])
print(A.T) # [[1 4] [2 5] [3 6]]
print(A.T.shape) # (3, 2)When shapes don't fit in a matrix multiplication, the fix is often to take the transpose of one side.
XᵀX and the normal equation
The closed-form solution of linear regression is written with the transpose: w = (XᵀX)⁻¹ Xᵀy.
X = np.array([[1, 1], [1, 2], [1, 3]]) # first column: 1 for the intercept
y = np.array([3, 5, 7])
w = np.linalg.solve(X.T @ X, X.T @ y)
print(w) # [1. 2.] -> y = 1 + 2xnp.linalg.solve is more stable and faster than taking the inverse explicitly. scikit-learn's LinearRegression also solves it in a similar way.
Practice Normal equation
X with a first column of 1s and y are given. Find w with np.linalg.solve(X.T @ X, X.T @ y) and print it.
Starter code
import numpy as np
X = np.array([[1, 0], [1, 1], [1, 2], [1, 3]])
y = np.array([4, 7, 10, 13])
# buraya yazSample solution
import numpy as np
X = np.array([[1, 0], [1, 1], [1, 2], [1, 3]])
y = np.array([4, 7, 10, 13])
w = np.linalg.solve(X.T @ X, X.T @ y)
print(w)Expected output
[4. 3.]
Questions and answers
Review the lesson questions together with their answers. To test yourself, solve the lesson interactively.
1 For X of shape (5, 3), what is the shape of <code>X.T @ X</code>?
- An error
- (3, 3) Correct answer
- (5, 5)
- (5, 3)
Explanation: (3, 5) @ (5, 3) = (3, 3); it is a matrix describing the relationships between features.
2 Take the transpose of the matrix.
At = A.____At = A.TExplanation: .T is the same as numpy.transpose(A).
3 Match the expressions.
| Term | Meaning |
|---|---|
A.T |
Transpose |
X.T @ X |
Feature × feature matrix |
np.linalg.solve |
Linear system solution |
(A.T).T |
A itself |