Question

State the Fundamental Subspace Theorem and give an example of its application in the case of a 2×2 matrix.

Answer 1

The fundamental theorem of linear algebra relates to the four fundamental subspaces associated with any mxn matrix A, namely col(A) and col(A^T), the column spaces of A and A^T respectively and Null(A) and Null(A^T), the null spaces of A and A^T respectively.

1. dim(Row(A)) = dim(Row(A^T))
2. dim(Row(A)) +dim(Null(A)) = n, the number of columns in A

       3. Null(A) ⊥ Row(A^T) i.e. Col (A^T) = Null(A) ^⊥ and Null(A^T) = Col(A) ^⊥ (Orthogonal complements in Rⁿ and R^m respectively)

4. Orthonormal base exist for both, the column space of A, i.e. col(A) and the row space of A i.e. col(A^T)

5. A is diagonal with respect to with respect to the orthonormal bases for col(A) and col(A^T)

When m = n = 2, i.e. when A is a 2x2 matrix, then all four of the fundamental matrix subspaces are lines in R².