Math 218: Matrices and Vectors

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Null Spaces

To check if a vector v is in the null space of a matrix A, we need only check that A*v is equal to the zero vector.

More efficiently, we can directly verify if v is in the null space of A with the syntax A*v == zero_vector(A.nrows()).

To check if v is an eigenvector of A corresponding to an eigenvalue l, we need to check if v is in the null space of l*identity_matrix(A.nrows())-A.

More efficiently, we can check if A*v == l*v.

For a 2x2 matrix, we can visualize the "input" vector v and the "output" vector A*v. The vector v is an eigenvector of A if input is parallel to the output.