When is the linear system Ax=b consistent?
Conditions
- A is a matrix representing a linear transformation.
- b is a vector with compatible dimensions.
Reasoning, step by step
- Identify the columns of matrix A.
- Determine the column space of A, which is the span of its columns.
- Check if the vector b can be written as a linear combination of the columns of A.
- Conclude that the system is consistent if and only if b is in the column space of A.
Example
The script states: 'The system Ax=b is consistent exactly when b lies in the column space of A.'
Common misconceptions
- Believing that a square matrix guarantees a solution for any b.
- Thinking that consistency depends solely on the number of rows and columns.
- Confusing the column space with the row space.
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