Linear Algebra Polynomials and Vectors Questions

I just need the True and False for these questions. No explanation needed. Thanks.

Linear Algebra
Final Exam – Page 3 of 8
2. (30 points) Determine whether or not the following statements are true or false.
(a) (3 points) The Pythagorean theorem ||$||2 + ||0||2 = ||u + v1|2 holds if and only if
u and v are orthogonal .
(b) (3 points) Given an m x n matrix A, Null(A) CR”.
(c) (3 points) Two eigenvectors corresponding to two distinct eigenvalues of an n xn
matrix A are linearly independent.
(d) (3 points) n x n matrices with the same characteristic polynomial are similar.
(e) (3 points) An n x n matrix A is invertible if and only if 0 is not an eigenvalue for
A.
(f) (3 points) An n x n matrix is invertible if and only if the transformation 2 H+ Ax
is an injective linear transformation R” +R”.
(g) (3 points) If A ~ B are row equivalent n x n matrices, then they have the same
eigenvalues.
(h) (3 points) A diagonalizable n x n matrix has n distinct eigenvalues.
(i) (3 points) Given a subspace W of a vector space V of dimension n we have dim(W)+
dim(W+) = n, where W+ is the orthogonal complement of W in V.
(j) (3 points) The map T(T) = Až +b is a linear transformation T:R” +R” for any
A E Mn(R), b E R”.