LAHW#11
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Transcript of 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 u1 = (1, 3, 2), u2 = (2, 1, 4), and u3 = (1, 0, 6).
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.)
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.
5.2 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.
71492
41131
31012
10121
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)).
121
253
132
231
5.2 Bases and Dimension
• 31.– Establish that if S is a linear independent set o
f n elements in a vector space, then Dim(Span(S)) = n. Every linear independent set is a basis for something.
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.
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.