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LAHW#11

LAHW#11. • Due December 6, 2010. 5.2 Bases and Dimension. 5. Find the coordinates of x = (-5, 1, 2) with respect to the basis consisting of u 1 = (1, 3, 2) , u 2 = (2, 1, 4) , and u 3 = (1, 0, 6). 5.2 Bases and Dimension. 9.

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LAHW#11

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  1. LAHW#11 •Due December 6, 2010

  2. 5.2 Bases and Dimension • 5. • Find the coordinates of x = (-5, 1, 2) with respect to the basis consisting of u1 = (1, 3, 2), u2 = (2, 1, 4), and u3 = (1, 0, 6).

  3. 5.2 Bases and Dimension • 9. • Explain why the map L: Pn → Pn defined by the equation L(p) = p + p’ is an isomorphism. You may assume that L is linear. (Here p’ it the derivative of p.)

  4. 5.2 Bases and Dimension • 10. • Define polynomials p1(t)=1–2t–t2, p2(t)=t+t2+t3, p3(t)=1–t+t3, and p4(t)=3+4t+t2+4t3. Let S be the set of these four functions. Find a subset of S that is a basis for the span of S.

  5. 5.2 Bases and Dimension • 22. • Let A = Find a simple basis for the column space of A. Explain why {x: Ax=0} is a subspace of R5, and compute its dimension.

  6. 5.2 Bases and Dimension • 29. • Let A = Find bases for the range and the kernel of A. Find values for Dim(Ker(A)), Dim(Range(A)), Dim(Domain(A)).

  7. 5.2 Bases and Dimension • 31. • Establish that if S is a linear independent set of n elements in a vector space, then Dim(Span(S)) = n. Every linear independent set is a basis for something.

  8. 5.2 Bases and Dimension • 34. • Verify that the dimension of Rm×n is mn. What is the dimension of the subspace of Rn×n consisting of symmetric matrices? An argument is required.

  9. 5.2 Bases and Dimension • 37. (Challenging) • Let B be an n×n noninvertible matrix. Let V be the set of all n×n matrices A such that BA = 0. Is V a vector space? If so, what is its dimension.

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