Math 108B - Home Work # 1
Due: Friday, April 11, 2008
1. Let T :R2 →R3 be the linear transformation given by the matrix
⎛
⎝ 1 −1
2 2
0 3
⎞
⎠
with respect to the standard bases. Find bases for R2 and R3 in which the matrix of
T is ⎛
⎝ 1 0 0 1 0 0
⎞
⎠
2. The matrix
4 −1
2 4
represents a linear transformationT :R2 →R2 with respect to the basis{v1, v2}where v1 = (1,1) and v2 = (−1,1). Find the matrix of T with respect to the basis {w1, w2} where w1 = (1,2) and w2 = (0,1).
3. Let T :V →W be a linear transformation, and let{v1, . . . , vn}be a basis forV. Show that T is invertible if and only if {T v1, . . . , T vn} is a basis for W.
4. The trace of an n×n matrix A is defined as the sum of all the entries on the main diagonal of A. That is,
tr(A) = n
i=1
Aii,
where Aij denotes the entry ofA in the ith row and jth column.
(a) Show that for any two n×n matrices A and B,tr(AB) = tr(BA).
(b) Use (a) to show that if X and Y are similar matrices then tr(X) = tr(Y).
5. LetV be an inner-product space, and letW be a subspace ofV. Define theorthogonal complement of W by
W⊥ ={v ∈V | v, w= 0 ∀w∈W}.
Show thatW⊥ is a subspace ofV.
1
Delhi School of Economics
Introductory Math Econ.
Problem Sets - Part A
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Instructor: SUGATA BAG
Delhi School of Economics
Introductory Math Econ.
Problem Sets - Part B
Instructor: SUGATA BAG
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Page | 2
7. Let x1 ,………, xn be positive real numbers. Prove that (Hint: Use the Cauchy- Schwarz inequality) –
Delhi School of Economics
Introductory Math Econ Problem Sets - Part C
Instructor: SUGATA BAG
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Page | 3 1. On P2(R), consider the inner product given by
Apply the Gram-Schmidt procedure to the basis (1, x, x2) to produce an orthonormal basis of P2(R).
2. Find an orthonormal basis of P2(R) (with inner product as in Exercise 1) such that the differentiation operator (the operator that takes p to pƍ ) on P2(R) has an upper-triangular matrix with respect to this basis.
3. Suppose U is a subspace of V. Prove that dimU٣= dimV dim U.
4. In R4, let U = span {(1, 1, 0, 0), (1, 1, 1, 2)}.
Find u א U such that || u (1, 2, 3, 4)|| is as small as possible.
5. (Do not turn in) For Exercise 1, does anything change if you apply the Gram- Schmidt Process to the basis {1, x, x2} for P2(C) with the inner product
1
, 0 ( ) ( ) p q
³
p x q x d x ?6. If U is a subset of an inner product space V (but not necessarily a subspace), we can still define –
