Facts for Exam #2
Math 2270, Spring 2005
Prove the following facts:
(1) Any two basis in a linear space have the same length.
(2) If W ⊂ V are two linear spaces, then dimW ≤dimV .
(3) If B is a basis of a linear space V , then the B-coordinate transformation is an isomorphism.
(4) If ~u1 , ~u2, . . . , ~um are orthonormal vectors in Rn , then they are linearly independent.
(5) If ~u1 , ~u2, . . . , ~un are orthonormal vectors in Rn , then they form a basis.
(6) Pythagorean theorem.
(7) Cauchy-Schwartz theorem.
(8) Let T : Rn → Rn be an orthogonal transformation, then for all ~x and ~y in Rn
~x ⊥ ~y =⇒ T (~x) ⊥ T (~y ).
(9) T : Rn → Rn is orthogonal if and only if {T (~e1 ), T (~e2 ), . . . , T (~en )} orthonormal.
(10) An n × n matrix A is orthogonal if and only if the columns form an orthonormal basis of Rn .
(11) An n × n matrix A is orthogonal if and only if AT A = In .
(12) For any matrix A, (imA)T = ker(AT ).
(13) If A is an n × m matrix, then ker(A) = ker(AT A).