Numerical Linear Algebra. Neuman Lemma.

Prove the Neuman Lemma: if ||A|| < 1, then I - A is invertible. Here ||.|| is a norm on the space of nxn matrices induced by a norm on R^n or C^n ( C is complex plane). I want a detailed proof. Thanks.

Subspace of inner product. Numerical Linear Algebra.

Please help with problem # 1 of this set. Thanks. (See attached file for full problem description)

Numerical Linear algebra. Orthogonal matrix.

Please help with problem 6 of this set. Don't need to describe the action of Q on R^2. (See attached file for full problem description)

Numerical Linear Algebra. Orthogonal matrix/eigenvalues.

Let A be a 3 x 3 real orthogonal matrix with det A = 1. Prove that lambda = 1 is an eigenvalue of A.

Numerical Linear algebra. Gram-Schmidt orthogonalization.

Please help with problem 5 to obtain orthogonal basis. Don't worry about the extra credit part. (See attached file for full problem description)