Linear Algebra

A steady-state vector for a stochastic matrix is actually an eigenvector.

TRUE A steady state vector has the property that Axx = x. In this case λ is 1.

If B is an echelon form of A, and if B has three nonzero rows, then the first three rows of A form a basis of Row A.

FALSE The nonzero rows of B form a basis. The first three rows of A may be linear dependent.

If Ax = λx for some scalar λ, then x is an eigenvector of A.

FALSE The vector must be nonzero.

The eigenvalues of a matrix are on its main diagonal.

FALSE This is only true for triangular matrices.

If Ax = λx for some vector x, then λ is an eigenvalue of A.

FALSE This is true as long as the vector is not the zero vector.

The dimensions of the row space and the column space of A are the same, even if A if A is not square.

TRUE by the Rank Theorem. Also since dimension of row space = number of nonzero rows in echelon form = number pivot columns = dimension of column space.

A matrix A is not invertible if and only if 0 is an eigenvalue of A.

TRUE.

If A and B are row equivalent, then their row spaces are the same.

TRUE. This allows us to find row space of A by finding the row space of its echelon form.

The sum of the dimensions of the row space and the null space of A equals the number of rows in A.

FALSE Equals number of columns by rank theorem. Also dimension of row space = number pivot columns, dimension of null space = number of non-pivot columns (free variables) so these add to total number of columns.

Row operations preserve the linear dependence relations among the rows of A.

FALSE For example, Row interchanges mess things up.

If B is any echelon form of A, the the pivot columns of B form a basis for the column space of A.

FALSE It's the corresponding columns in A.

To find the eigenvalues of A, reduce A to echelon form.

FALSE Row reducing changes the eigenvectors and eigenvalues.

If v1 and v2 are linearly independent eigenvectors, then they correspond to different eigenvalues.

FALSE The converse if true, however.

The row space of A^T is the same as the column space of A.

TRUE Columns of A go to rows of A^T.

On a computer, row operations can change the apparent rank of a matrix.

TRUE Due to rounding error.

Finding an eigenvector of A may be difficult, but checking whether a given vector is in fact an eigenvector is easy.

TRUE Just see if Ax is a scalar multiple of x.

An eigenspace of A is a null space of a certain matrix.

TRUE The eigenspace is the nullspace of A − λI.

The row space of A is the same as the column space of A^T.

TRUE The rows become the columns of A^T so this makes sense.

The dimension of null space of A is the number of columns of A that are not pivot columns.

TRUE These correspond with the free variables.

A number c is an eigenvalue of A if and only if the equation (A−cI)x=0 has a nontrivial solution.

TRUE This is a rearrangement of the equation Ax = λx.