np.linalg.solve() for linear systems in NumPy - Step-by-Step Execution

Concept Flow - np.linalg.solve() for linear systems

Start with matrix A and vector b

↓

Check if A is square and dimensions match b

Yes↓

Use np.linalg.solve(A, b) to find x

↓

Return solution vector x

↓

End

The function takes a square matrix A and vector b, checks dimensions, then solves Ax = b for x.

Execution Sample

NumPy

import numpy as np
A = np.array([[3,1],[1,2]])
b = np.array([9,8])
x = np.linalg.solve(A, b)
print(x)

Solves the system 3x + y = 9 and x + 2y = 8 for x and y.

Execution Table

Step	Action	Input	Intermediate Result	Output
1	Define matrix A	[[3,1],[1,2]]	Matrix A created
2	Define vector b	[9,8]	Vector b created
3	Check dimensions	A shape (2,2), b shape (2,)	Dimensions valid
4	Call np.linalg.solve(A,b)	A, b	Computes solution x
5	Return solution x		[2. 3.]	x = [2. 3.]
6	Print x			[2. 3.]

💡 Solution found because A is square and dimensions match b

Variable Tracker

Variable	Start	After Step 1	After Step 2	After Step 4	Final
A	undefined	[[3 1] [1 2]]	[[3 1] [1 2]]	[[3 1] [1 2]]	[[3 1] [1 2]]
b	undefined	undefined	[9 8]	[9 8]	[9 8]
x	undefined	undefined	undefined	[2. 3.]	[2. 3.]

Key Moments - 3 Insights

Why must matrix A be square to use np.linalg.solve()?

np.linalg.solve() requires a square matrix because only square matrices have unique solutions for Ax = b. See execution_table step 3 where dimension check ensures A is square.

What happens if the dimensions of b do not match A?

The function will raise an error because b must have the same number of rows as A. This is checked in execution_table step 3.

Why does np.linalg.solve() return a vector x?

Because it solves the equation Ax = b for x, which is the vector of unknowns. This is shown in execution_table step 5 where the solution vector is returned.

Visual Quiz - 3 Questions

Look at the execution table, what is the value of x after step 4?

A [3. 1.]

B [2. 3.]

C [9. 8.]

D undefined

💡 Hint

Check the 'Intermediate Result' column at step 4 in execution_table.

At which step does the code check if A is square and matches b's dimensions?

A Step 2

B Step 4

C Step 3

D Step 5

💡 Hint

Look for the 'Check dimensions' action in execution_table.

If b had shape (3,) instead of (2,), what would happen?

A The function would raise a dimension mismatch error

B np.linalg.solve() would return a solution vector of length 3

C The function would ignore extra elements in b

D The function would solve with the first two elements only

💡 Hint

Refer to key_moments about dimension matching and execution_table step 3.

Concept Snapshot

np.linalg.solve(A, b)
- Solves Ax = b for x
- A must be square matrix
- b must match A's row count
- Returns solution vector x
- Fast and accurate for linear systems

Full Transcript

This visual trace shows how np.linalg.solve() works step-by-step. First, we define matrix A and vector b. Then, the function checks if A is square and if b's dimensions match. If valid, it computes the solution vector x that satisfies Ax = b. The solution is returned and printed. Key points include the requirement for A to be square and dimension matching with b. The execution table tracks each step and variable changes, helping beginners understand the process clearly.