Column Spaces and Null Spaces

Column Space

• Definition: The set of all linear combinations of the columns of matrix A.
  • Matrix A is composed of vectors.
  • The column space of A is the span of those vectors.
• Another Expression: The column space of A is equal to all b such that b = Ax for some x in Rn.
• Theorem: If A is an m by n matrix, the column space of A is a subspace of Rm.
  • Proof is often omitted as it follows from the properties of vector spaces and spans.

Practice Problems

• Example: Determining if a vector u is in the null space and/or column space.
  • Null Space Check:
    1. Multiply matrix A by vector u.
    2. If the result is the zero vector, then u is in the null space.
    3. Example Calculation:
    • A = [ [ 2, -6, 12 ], [ -3, 9, 9 ], [ 0, 0, 1 ] ]
    • u = [ 3, 1, 0 ]
    • Result: Au = [ 0, 0, 0 ] (u belongs to null A).
  • Column Space Check:
    1. Augment A with u.
    2. Perform row reduction to determine if the system is consistent.
    3. If consistent, u is in the column space. If inconsistent, u is not.
    4. Example Calculation:
    • Augmented matrix: [ [ 2, -6, 12, 3 ], [ -3, 9, 9, 1 ], [ 0, 0, 1, 0 ] ]
    • Perform row reductions to determine inconsistency.
    • Conclusion: System is inconsistent (u does not belong to column space of A).

Finding Specific Matrices and Vectors

• Finding a Matrix A:
  • Given: A set W that spans vectors.
  • Rewrite W as a linear combination of vectors.
  • Form matrix A such that W is equal to the column space of A.
  • Example:
    • W = span{ [4, 1, 0], [2, -1, -4] }
    • Matrix A = [ [4, 1, 0], [2, -1, -4] ]
• Non-Zero Vectors in Column and Null Space:
  • Column Space: Any column of A written as a vector.
  • Null Space: Find by row reducing A augmented with the zero vector. Identify free variables and construct a vector.
  • Example Calculation for Null Space:
    • A = [ [ 1, 0, 3, 4 ], [ 0, 1, 4, -1 ], [ 0, 2, 0, 6 ] ]
    • Perform row reductions.
    • Free variable solution: x1 = -7x4, x2 = -3x4, x3 = x4, x4 is free.
    • Example vector: Choose x4 = 2 -> [-14, -6, 2, 2]

Additional Questions

• Null Space Check Recap: Multiply A by u and compare result with zero vector.
• Column Space Check Recap:
  • Compare vector dimensions with dimensions expected in column space.
• Linear Transformations into Vector Spaces:
  • Definition: Assigns unique vector T(x) to each vector x.
  • Properties:
    1. T(u + v) = T(u) + T(v) for all u, v in vector space V.
    2. T(cu) = cT(u) for scalar c.
  • Terms:
    • Kernel: Null space of T (all vectors mapped to zero).
    • Range: Set of all vectors in W of the form T(x) for some x in V.