By Cohn P. M.

Stories very important heritage in algebra and introduces extra complex themes, emphasizing linear algebra and the homes of teams and jewelry. contains extra labored difficulties and a whole set of solutions to the routines. additionally good points multiplied proofs and extra in-depth remedies of affine areas, linear programming, duality, Jordan common shape and staff conception.

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C) Deduce that the rank of A is m − j . 11. 5. (a) Suppose y ∈ R(A)⊥ . This clearly implies Ax, y = 0 for all x ∈ Fm . 2 to show that A∗ y, A∗ y = 0, from which you can deduce that y ∈ N (A∗ ). This demonstrates that R(A)⊥ ⊆ N (A∗ ). (b) Now make an argument that is approximately the reverse of the argument of part (a) to deduce the reverse inclusion N (A∗ ) ⊆ R(A)⊥ . (c) Show that R(A∗ )⊥ = N (A). 12. 11. Let S, Sˆ ∈ Fn×k . ˆ ⊆ R(S). 6. Projectors ✐ 25 ˆ , where Y = X −1 . Deduce (b) Show that if Sˆ = SX and X is nonsingular, then S = SY ˆ ˆ that R(S) ⊆ R(S).

If we want to stay within the real number system, we should work with subspaces of Rn instead. 3. 5. Let A ∈ Rn×n with n ≥ 3. Then A has a nontrivial invariant subspace. Proof: Let λ be an eigenvalue of A. If λ is real, then let v ∈ Rn be a real eigenvector of A associated with λ. Then S 1 = span{v} is a one-dimensional (hence nontrivial) subspace of Rn that is invariant under A. Now suppose that λ is not real, and let v ∈ Cn be an eigenvector associated with λ. Then v cannot possibly be real. Write v = v1 + iv2 , where v1 , v2 ∈ Rn .

Sk form a basis for S, then each vector v ∈ S can be expressed as a linear combination of s1 , . . , sk in exactly one way. That is, for each v ∈ S, there are unique α1 , . . , αk ∈ F such that v = α1 s1 + · · · + αk sk . Recall that the set s1 , . . , sk is called orthonormal if si , sj = δij . Every orthonormal set is linearly independent. ) An orthonormal set that is also a basis of S is called an orthonormal basis of S. We will not work out the theory of bases and dimension, which can be found in any elementary linear algebra text.