In Problems 43–54, calculate the definite integral, given that
∫
1
4
x
d
x
=
7.5
∫
1
4
x
2
d
x
=
21
∫
4
5
x
2
d
x
=
61
3
49.
∫
1
5
6
x
2
d
x
With differentiation, one of the major concepts of calculus. Integration involves the calculation of an integral, which is useful to find many quantities such as areas, volumes, and displacement.
1. True or false:
(a) if E is a subspace of V, then dim(E) + dim(E+) = dim(V)
(b) Let {i, n} be a basis of the vector space V, where vi,..., are all eigen-
vectors for both the matrix A and the matrix B. Then, any eigenvector of A is
an eigenvector of B.
Justify.
2. Apply Gram-Schmidt orthogonalization to the system of vectors {(1, 2, -2), (1, −1, 4), (2, 1, 1)}.
3. Suppose P is the orthogonal projection onto a subspace E, and Q is the orthogonal
projection onto the orthogonal complement E.
(a) The combinations of projections P+Q and PQ correspond to well-known oper-
ators. What are they? Justify your answer.
(b) Show that P - Q is its own inverse.
4. Show that the Frobenius product on n x n-matrices,
(A, B) =
= Tr(B*A),
is an inner product, where B* denotes the Hermitian adjoint of B.
5. Show that if A and B are two n x n-matrices for which {1,..., n} is a basis of eigen-
vectors (for both A and B), then AB = BA.
Remark: It is also true that if AB = BA, then there exists a common…
Question 1. Let f: XY and g: Y Z be two functions. Prove that
(1) if go f is injective, then f is injective;
(2) if go f is surjective, then g is surjective.
Question 2. Prove or disprove:
(1) The set X = {k € Z} is countable.
(2) The set X = {k EZ,nЄN} is countable.
(3) The set X = R\Q = {x ER2
countable.
Q} (the set of all irrational numbers) is
(4) The set X = {p.√2pQ} is countable.
(5) The interval X = [0,1] is countable.
Question 3. Let X = {f|f: N→ N}, the set of all functions from N to N. Prove
that X is uncountable.
Extra practice (not to be submitted).
Question. Prove the following by induction.
(1) For any nЄN, 1+3+5++2n-1 n².
(2) For any nЄ N, 1+2+3++ n = n(n+1).
Question. Write explicitly a function f: Nx N N which is bijective.
3. Suppose P is the orthogonal projection onto a subspace E, and Q is the orthogonal
projection onto the orthogonal complement E.
(a) The combinations of projections P+Q and PQ correspond to well-known oper-
ators. What are they? Justify your answer.
(b) Show that P - Q is its own inverse.
Chapter 5 Solutions
Calculus for Business, Economics, Life Sciences, and Social Sciences (13th Edition)
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Definite Integral Calculus Examples, Integration - Basic Introduction, Practice Problems; Author: The Organic Chemistry Tutor;https://www.youtube.com/watch?v=rCWOdfQ3cwQ;License: Standard YouTube License, CC-BY