Question

Prove that rowspace A = (nullspace A) and colspace A = (nullspace Aᵀ)⊥.

Suggestion: First show rowspace A ⊆ (nullspace A)⊥, and then compare dimensions (implicit in Lemma 3.7 is the fact that for a suspace M of Rⁿ, dim M + dim M⊥ = n (justify); we also have the rank-nullity theorem).

Answer 1

To prove the relationship between the rowspace, colspace, and nullspaces of a matrix, we use the orthogonal relationship between vectors and the rank-nullity theorem, demonstrating that rowspace A is the orthogonal complement of nullspace A, and colspace A is the orthogonal complement of nullspace Aᵀ.

Step-by-step explanation:

To prove that the rowspace of a matrix A is the orthogonal complement of the nullspace of A, and the colspace of A is the orthogonal complement of the nullspace of Aᵀ, consider the following steps:

• First, to show that rowspace A ⊆ (nullspace A)⊥, note that if a vector v is in the nullspace of A, then Av = 0. This means that v is orthogonal to every row of A, and thus it is in the orthogonal complement of rowspace A.
• Using the rank-nullity theorem, which states that the rank of A plus the nullity of A equals the number of columns n, we can deduce that dim(rowspace A) + dim(nullspace A) = n.
• By the property that the dimension of a subspace plus the dimension of its orthogonal complement equals the whole space, it follows that rowspace A = (nullspace A)⊥.
• Similarly, since Aᵀ has the same null vectors as A and the rows of Aᵀ correspond to the columns of A, we can say that colspace A = (nullspace Aᵀ)⊥.