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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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- Find a basis for R3 that includes the vector (1,0,2) and (0,1,1).arrow_forwardIn Exercises 11–12, use the Subspace Test to determine which of the sets are subspaces of M22. а 11. a. All matrices of the form b l: a 1 b. All matrices of the form b 1 c. All 2 × 2 matrices A such that 2 Аarrow_forwardExercises 1. Show that {(1,1,0,0), (1,0, 1, 1)} and {(2,-1,3,3), (0,1,-1,-1)} subspace of R4. span the samearrow_forward
- 5. Find the dimension of the subspace spanned by the given vectors: u₁ = (1, 3, -2,5), U₂ = (0, 1,-1, 2), u3 = (2, 1, 1, 2).arrow_forwardDetermine whether the sets defined by the following vectors are subspaces of R3. (a) (a,b,2a + 3b) (c) (a,a+2,b) (b) (a, b, 3) (d) (a, -a, 0)arrow_forward9. Determine whether the vectors v₁ = (1,1,2), v₂ = (1,0,1) and v3 = (2,1,3) span the vector space R³.arrow_forward
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- If 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_forwardExercises 9–12 display a matrix and an echelon form of . Find bases for Col and Nul , and then state the dimensions of these subspaces.arrow_forward
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