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Linear Algebra: Null Space and Column Space

I have attached a word document that contains my question. In the attached document R( ) is the row space, N( ) is the null space, and C( ) is the column space.

If you were to let A be a 6 x 14 matrix where the dimension of the row space is 3 (dim(R(A) = 3), what would the dimension of the null space of matrix A (dim(N(A)) be and what would the dimension of the null space of A^T (dim(N(A^T)) be?

Also, if you let B be a 6 x 6 invertible matrix, what would dim(N(BA)^orthogonal complement) and dim(C(A^TB^T)^orthogonal complement) be? (C( ) is the column space)

Solution Summary

This solution is provided within a Word document which is attached and discusses concepts including the orthogonal complement, null space and column space. The solution is detailed, well presented and includes all calculations pertaining to the problem.