2.1-03 The column picture

The column picture of a linear system

Learn to see a linear system as a target vector assembled from weighted coefficient columns. This questionset in Module 2, Chapter 1 of the Complete Linear Algebra Course develops the connection between simultaneous equations, vector addition and scalar multiplication.

What you will practise

• Extract columns from a system while preserving equation and variable order.
• Compute weighted sums and recover coefficients by matching coordinates.
• Interpret systems with different numbers of equations and unknowns.
• Decide whether a target is reachable from the available column directions.
• Recognize when different weights give the same target, including zero or dependent columns.
• Distinguish unrestricted real weights from additional nonnegative-weight conditions.
• Resolve a parameter-dependent target and transfer weight between related columns.

Early questions build fluency with short vectors. Later questions ask you to justify existence, uniqueness or multiple representations without requiring a formal elimination algorithm. Single-select and multi-select formats encourage both calculation and careful evaluation of conceptual claims. Explanations work coordinate by coordinate and connect the result back to the vector picture.

Before you begin

You should know vector coordinates, scalar multiplication, vector addition and the introductory meaning of linear independence from Module 1. Familiarity with the row picture will help you compare the two interpretations. The focus here is on understanding what the weights mean and why a proposed combination succeeds or fails.