Solving linear systems (solve) in SciPy - Step-by-Step Execution

Concept Flow - Solving linear systems (solve)

Define matrix A and vector b

↓

Call solve(A, b) function

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Check if A is square and invertible

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Compute solution x

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Return solution x

We start with matrix A and vector b, then call solve. It checks if A can be inverted, then computes and returns the solution vector x.

Execution Sample

SciPy

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

This code solves the system Ax = b for x using scipy's solve function.

Execution Table

Step	Action	Evaluation	Result
1	Define matrix A	A = [[3,1],[1,2]]	Matrix A created
2	Define vector b	b = [9,8]	Vector b created
3	Call solve(A, b)	Check if A is square and invertible	A is 2x2 and invertible
4	Compute solution x	Solve Ax = b	x = [2. 3.]
5	Print x	Output solution vector	[2. 3.]

💡 Solution found because A is square and invertible

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 solve?

Because solve requires a square matrix to find a unique solution. The execution_table step 3 shows the check for squareness before computing x.

What happens if matrix A is not invertible?

solve will raise an error because no unique solution exists. This is implied in step 3 where invertibility is checked before solving.

Why is the solution vector x a 1D array?

Because b is a 1D vector and solve returns x with the same shape, representing the solution values for each variable, as shown in step 4 and 5.

Visual Quiz - 3 Questions

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

A [3, 1]

B [9, 8]

C [2, 3]

D [1, 2]

💡 Hint

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

At which step does the code check if matrix A is invertible?

A Step 2

B Step 3

C Step 4

D Step 5

💡 Hint

Look at the 'Action' and 'Evaluation' columns in the execution_table.

If vector b was changed to [10, 5], what would change in the variable_tracker?

A Only b and x values would change

B Only A would change

C A, b, and x would all change

D No variables would change

💡 Hint

Refer to variable_tracker rows for b and x values.

Concept Snapshot

Use scipy.linalg.solve(A, b) to find x in Ax = b
- A must be a square, invertible matrix
- b is the right-hand side vector or matrix
- Returns x, the solution vector
- Raises error if no unique solution
- Efficient and reliable for linear systems

Full Transcript

We start by defining matrix A and vector b representing the system Ax = b. Then we call scipy.linalg.solve(A, b). The function first checks if A is square and invertible. If yes, it computes the solution vector x that satisfies the equation. Finally, x is returned and printed. If A is not invertible, solve raises an error. The solution vector x matches the shape of b and contains the values for the variables that solve the system.