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.