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Linear Algebra
Show that the ith column of A^(conjugate transpose) is normal to the hyperplane Hi; that is, it is orthogonal to every other vector lying in Hi.
Linear Algebra
Let A be an N by N matrix with rank J<N. Prove then, that there are N-J linearly dependent solutions of the system Ax = 0, and the null space of A has dimension N-J.
Linear Algebra
Prove these assertions concerning the effects of row operations on the determinant:
1) multiplying one row by k multiplies the determinant by k.
2) interchanging two rows changes the sign of the determinant.
3) adding to one row a multiple of another row has no effect on the determinant.
1) multiplying one row by k multiplies the determinant by k.
2) interchanging two rows changes the sign of the determinant.
3) adding to one row a multiple of another row has no effect on the determinant.
Linear Algebra
Let A be an N by N matrix with rank J<N. Prove then that there are N - J linearly independent solutions of the system Ax = 0, and the null space of A has dimension N - J.
Linear Algebra
Let W = {w^1,....,w^N} be a spanning set for a subspace W in R^K, and U = {u^1,....,u^M} a linearly independent subset of W. Let A be the K by M matrix whose columns are the vectors u^m, and B the K by N matrix whose column vectors are w^n. Then there is an N by M matrix D such that A = BD. Why??
(hint: prove by showing that, if M>N, then there is A non-zero vector x with Dx = 0.)
(hint: prove by showing that, if M>N, then there is A non-zero vector x with Dx = 0.)
Linear Algebra
Let A be an arbitrary M by N matrix and B the matrix in row-reduced echelon form obtained from A. Prove that there is a non-zero solution of the system of linear equations Ax = 0 iff B has fewer than N non-zero rows.
Linear Algebra
Prove that if A is M by N and M<N, then there is a non-zero x with Ax = 0.
Linear Algebra
Prove that any matrix A can be transformed into a matrix B in row-reduced echelon form using elementary row operations.
Linear Algebra
Research the Cayley-Hamilton Theorem and how it can be used to compute the inverse of a non-singular square matrix.
Linear Algebra
Why do we impose the requirement that the eigenvector be non-zero when we do not place this requirement on the eigenvalue.
