Linear Algebra Problems and Proofs in Subspaces
Explore various proofs and problems related to subspaces in linear algebra, including orthogonal bases, unique vector expressions, and matrix properties. Learn about bases for subspaces, proving uniqueness, and orthogonal vector spaces.
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7.3 S is the set of vectors that are orthogonal to every vector in S ? = ?:? ? = 0, ? ?
w ? = ?1?1+ ?2?2+ + ???? +?1?1+ ?2?2+ + ?? ??? ? Unique z ???? + ???? = ? For any subspace W of Rn Basis: ?1,?2, ,?? Basis: ?1,?2, ,?? ? Basis for Rn For every vector u, W u u = w + z (unique) z W w ? ? 0
58. Let ? be a subspace of ??, and let B1 and B2 be bases for ? and ?, respectively. (a) ?1 ?2 is a basis for ?? (b) ???? + ???? = ? 4-2: 77 Let ? and ? be nonzero subspaces of ?? such that each vector ? in ??can be uniquely expressed in the form ? = ? + ?for some ? in ? and some ? in W. (a) Prove that ?is the only vector in both ? and ?. (b) Prove that ???? + ???? = ?
61. Prove the following statements for any matrix ?: (a) ??? ? = ???? ? (b) ??? ? = ???? ?? 65. Let ? be an ? ? matrix. Prove that if ? is a vector in both ???A and ?????, then ? = 0.
57. Let ? be a nonempty finite subset of ??, and suppose that ? = ???? ?. Prove that ? = ? .
60. Prove that for any subspace ? of ??, ? = ? 63. Prove that for any nonempty finite subset ? of ??, ? = ???? ? 57. Let ? be a nonempty finite subset of ??, and suppose that ? = ???? ?. Prove that ? = ? .
64. Use the fact the ??? ?= ???? ? for any matrix ? to give another proof that ???? + ???? = ? for any subspace ? of ??. Hint: Let ? be a ? ? matrix whose rows constitute a basis for ?. W = ??? ?
59. Suppose that ?1,?2,,?? is an orthogonal basis for ??. For any ?, where 1 ? < ?, define ? = ???? ?1,?2, ,??. Prove that ??+1,??+2, ,?? is an orthogonal basis for ? .
67. Let ? be a subspace of ??. 2= ?? (a) Prove that ?? ?= ?? (b) Prove that ??
67. Let ? be a subspace of ??. 2= ?? (a) Prove that ?? ?= ?? (b) Prove that ??
