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Four Fundamental Subspaces Visualizer

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About This MicroSim

This visualization illustrates the Fundamental Theorem of Linear Algebra, showing the four fundamental subspaces associated with any matrix A (m x n):

Domain (R^n)

  • Row Space: The span of the rows of A (dimension = rank)
  • Null Space: Vectors x where Ax = 0 (dimension = n - rank)

Codomain (R^m)

  • Column Space: The span of the columns of A (dimension = rank)
  • Left Null Space: Vectors y where A^T y = 0 (dimension = m - rank)

Key Relationships

The four subspaces satisfy these fundamental properties:

\[\text{Row Space} \perp \text{Null Space}\]
\[\text{Column Space} \perp \text{Left Null Space}\]
\[\dim(\text{Row Space}) + \dim(\text{Null Space}) = n\]
\[\dim(\text{Column Space}) + \dim(\text{Left Null Space}) = m\]

The transformation A maps the Row Space onto the Column Space, and maps the Null Space to the zero vector.

How to Use

  1. Edit the matrix A using the input fields (up to 4x4 matrix)
  2. Click "Compute Subspaces" to calculate and display all four subspaces
  3. Toggle "Show Basis Vectors" to see the basis for each subspace
  4. Toggle "Verify Orthogonality" to check that orthogonal pairs have zero dot product
  5. Use the Highlight slider to focus on one subspace at a time:
  6. 0 = All subspaces
  7. 1 = Row Space
  8. 2 = Column Space
  9. 3 = Null Space
  10. 4 = Left Null Space

Key Observations

  • Rank determines all dimensions: Once you know rank(A), you know the dimension of all four subspaces
  • Orthogonal complements: Row space and null space are orthogonal complements in R^n; column space and left null space are orthogonal complements in R^m
  • Rank-deficient matrices: When rank < min(m,n), the null space and/or left null space are nontrivial
  • Full rank: When rank = min(m,n), either null space or left null space reduces to {0}

Embedding

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<iframe src="https://dmccreary.github.io/linear-algebra/sims/four-subspaces/main.html" height="542px" scrolling="no"></iframe>

Lesson Plan

Learning Objectives

Students will be able to:

  1. Identify and describe all four fundamental subspaces of a matrix
  2. Verify the orthogonality relationships between subspace pairs
  3. Apply the rank-nullity theorem to determine subspace dimensions
  4. Explain how the transformation A relates the domain and codomain subspaces

Suggested Activities

  1. Verify dimensions: For various matrices, confirm rank + nullity = n
  2. Orthogonality test: Pick vectors from row space and null space, verify dot product = 0
  3. Mapping exploration: Trace how a row space vector maps to column space, and how a null space vector maps to zero
  4. Rank variations: Compare subspaces for full rank vs rank-deficient matrices

Assessment Questions

  1. If a 3x4 matrix has rank 2, what are the dimensions of all four subspaces?
  2. Why must the row space and null space be orthogonal?
  3. If Ax = b has a solution, what does this say about b and the column space?
  4. How does the left null space relate to the solvability of A^T y = c?

References

  • Chapter 8: Vector Spaces and Subspaces
  • Strang, G. "The Fundamental Theorem of Linear Algebra"
  • Rank-Nullity Theorem