Math 218: Matrices and Vectors

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The Four Fundamental Subspaces

To verify if v is in the null space of a matrix A, we can use the syntax A*v == zero_vector(A.nrows()).

To verify if v is in the left null space of a matrix A, we can use the syntax A.T*v == zero_vector(A.ncols()).

To verify if v is in the column space of a matrix A, we can use the syntax A.rank() == A.augment(v).rank().

To verify if v is in the row space of a matrix A, we can use the syntax A.T.rank() == A.T.augment(v).rank().