By Gert Kjaergard Pedersen
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PROOF OF THEOREM 4 As was pointed out after Theorem 4, statements (a), (b), and (c) are logically equivalent. So, it sufﬁces to show (for an arbitrary matrix A) that (a) and (d) are either both true or both false. This will tie all four statements together. Let U be an echelon form of A. Given b in Rm , we can row reduce the augmented matrix Œ A b to an augmented matrix Œ U d for some d in Rm : ŒA b ŒU d If statement (d) is true, then each row of U contains a pivot position and there can be no pivot in the augmented column.
E. The set Span fu; vg is always visualized as a plane through the origin. 24. a. Any list of ﬁve real numbers is a vector in R5 . b. The vector u results when a vector u vector v. v is added to the c. The weights c1 ; : : : ; cp in a linear combination c1 v1 C C cp vp cannot all be zero. d. When u and v are nonzero vectors, Span fu; vg contains the line through u and the origin. e. Asking whether the linear system corresponding to an augmented matrix Œ a1 a2 a3 b has a solution amounts to asking whether b is in Span fa1 ; a2 ; a3 g.
2 3 2 3 7 5 15. v1 D 4 1 5 ; v2 D 4 3 5 6 0 2 3 2 3 3 2 16. 3 2 3 2 3 2 3 1 2 4 17. Let a1 D 4 4 5, a2 D 4 3 5, and b D 4 1 5. For what 2 7 h value(s) of h is b in the plane spanned by a1 and a2 ? 2 3 2 3 2 3 1 3 h 18. Let v1 D 4 0 5, v2 D 4 1 5, and y D 4 5 5. For what 2 8 3 value(s) of h is y in the plane generated by v1 and v2 ? 19. Give a geometric description of Span fv1 ; v2 g for the vectors 2 3 2 3 8 12 v1 D 4 2 5 and v2 D 4 3 5. 6 9 20. Give a geometric description of Span fv1 ; v2 g for the vectors in Exercise 16.
C*-algebras and their automorphism groups by Gert Kjaergard Pedersen