Linear Algebra

The Four Fundamental Subspaces

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Session Overview

For some vectors b the equation Ax = b has solutions and for others it does not. Some vectors x are solutions to the equation Ax = 0 and some are not. To understand these equations we study the column space, nullspace, row space and left nullspace of the matrix A.

Session Activities

Lecture Video and Summary

• Watch the video lecture The Four Fundamental Subspaces
• Read the accompanying lecture summary (PDF)
• Lecture video transcript (PDF)

Suggested Reading

• Read Section 3.6 in the 4^th edition or Section 3.5 in the 5^th edition.

Problem Solving Video

• Watch the recitation video on Computing the Four Fundamental Subspaces
• Recitation video transcript (PDF)

Check Yourself

Problems and Solutions

Work the problems on your own and check your answers when you’re done.

• Problems (PDF)
• Solutions (PDF)

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