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In Exercises 7–10, let W be the subspace spanned by the u’s, and write y as the sum of a
7. y =
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Linear Algebra and Its Applications (5th Edition)
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- Suppose C is a linear code (i.e. subspace), and u,v are in the same coset of C. Show u+C = v+Carrow_forwardIf U and W are subspaces of V such that V = U + W and Un W= {0}, then prove that every vector in V has a unique representation of the form u+ w where u is in U and w is in W. For the toolbar, press ALT+F10 (PC) or ALT+FN+F10 (Mac). U Paragraph Arial 14px Aarrow_forward3. Is the set of all vectors (x, y) in R2 with the usual addition and scalar multiplication, a subspace of R2? Justify your answer.arrow_forward
- 6. Let V be a vector space and v1, v2, V3, V4 are vectors in V. Suppose that {v1, v2, V3} are linear dependent and {v2, v3, V4} are linear independent. Show that (i) vị is a linear combination of v2, V3, and (ii) v¼ is not a linear combination of v1, v2, V3.arrow_forwardLet 2 3 2 X = X2 = X3 = 6. 3 4 4 (a) Show that x1, X2, and x3 are linearly dependent. (b) Show that x¡ and x2 are linearly independent. (c) What is the dimension of Span(x1, X2, X3)?arrow_forwardConsider V = R^3 and U consists of those vectors each of whose first component is non-negative. [i.e., U = {(a, b, c) | a ≥ 0 }. Answer with proper reasoning whether or not U is a subspace of Varrow_forward
- АT (Ь — Ах) — ӧ. Interpret this statement in terms of the fundamental subspaces of A.arrow_forwardSection 4.5 The Dimension of a Vector Space Prereqs: Basis (4.3), coordinate vector and coordinate mapping (4.4). (8.1) ** - Warm Up 1. Explain why B = is a basis for R².arrow_forward2: Let L be the subspace of V that contains all the real n x n lower triangular matrices. What is the dimension of L? You answer should not require more than 4 sentences.arrow_forward
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:CengageElementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage Learning
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