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= Let C be a set of all complex numbers (i.e., numbers of the form z = a +ib, where a and b are real numbers and i is imaginary unit, which satisfies į² -1). Let + be the usual addition of complex numbers (i.e., if z = a + ib and w= a' + ib', then z+w= (a + a') + i(b + b′)), and let be the usual multiplication of complex numbers (i.e., if z = a + ib and w = a' + ib', then zw= aa' — bb' + i(ba' + ab')). Check that C is a complex vector space (the definition of the complex vector space is identical to the real vector space given in Problem 2, except that the real scalars c, d there are replaced by the complex scalars, i.e., c, d is now in C, too). For this, verify that all properties of the vector space are satisfied.

= Let C be a set of all complex numbers (i.e., numbers of the form z = a +ib, where a and b are real numbers and i is imaginary unit, which satisfies į² -1). Let + be the usual addition of complex numbers (i.e., if z = a + ib and w= a' + ib', then z+w= (a + a') + i(b + b′)), and let be the usual multiplication of complex numbers (i.e., if z = a + ib and w = a' + ib', then zw= aa' — bb' + i(ba' + ab')). Check that C is a complex vector space (the definition of the complex vector space is identical to the real vector space given in Problem 2, except that the real scalars c, d there are replaced by the complex scalars, i.e., c, d is now in C, too). For this, verify that all properties of the vector space are satisfied.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Let C be a set of all complex numbers (i.e., numbers of the form z = a- a+ib, where a and b are real numbers and i is imaginary unit, which satisfies i² -1). Let + be the usual addition of complex numbers (i.e., if z = a + ib and w = a' + ib', then z + w = (a + a') + i(b + b′)), - and let be the usual multiplication of complex numbers (i.e., if z = a + ib and w = a' + ib', then zw= = aa' — bb' + i(ba' + ab')). Check that C is a complex vector space (the definition of the complex vector space is identical to the real vector space given in Problem 2, except that the real scalars c, d there are replaced by the complex scalars, i.e., c, d is now in C, too). For this, verify that all properties of the vector space are satisfied. .
Transcribed Image Text:Let C be a set of all complex numbers (i.e., numbers of the form z = a- a+ib, where a and b are real numbers and i is imaginary unit, which satisfies i² -1). Let + be the usual addition of complex numbers (i.e., if z = a + ib and w = a' + ib', then z + w = (a + a') + i(b + b′)), - and let be the usual multiplication of complex numbers (i.e., if z = a + ib and w = a' + ib', then zw= = aa' — bb' + i(ba' + ab')). Check that C is a complex vector space (the definition of the complex vector space is identical to the real vector space given in Problem 2, except that the real scalars c, d there are replaced by the complex scalars, i.e., c, d is now in C, too). For this, verify that all properties of the vector space are satisfied. .
Definition: A real vector space is a set V on which two operations + and (called vec- tor addition and scalar multiplication) are defined so that for each pair of elements {x,y} in V there is a unique element u+v in V, and for each element a in R (called a scalar) and each element x (called a vector) in V there is a unique element av in V, such that the following conditions hold. (1) Commutative law for addition: For all elements u and v in V, u + v = v + u (2) Associative law for addition: For all elements u, v, w in V, u + (v + w) = (u + v) + w (3) Additive identity: The set V contains an additive identity element, denoted by 0, such that for any element v in V, v + 0 = v. (4) Additive inverses: For each element u in V, the equation v + x = 0 has a solution x in V, called an additive inverse of v. (5) Unitary law: For all elements v in V, lv = v (6) Distributive law I: For all real numbers a and all elements u, v in V, c(u + v) = cu + cv (7) Distributive law II: For all real numbers c, d and all elements v in V, (c+d)v = cv + dv (8) Associative law for multiplication: For all real numbers c, d and all elements v in V, c(dv) = (cd)v
Transcribed Image Text:Definition: A real vector space is a set V on which two operations + and (called vec- tor addition and scalar multiplication) are defined so that for each pair of elements {x,y} in V there is a unique element u+v in V, and for each element a in R (called a scalar) and each element x (called a vector) in V there is a unique element av in V, such that the following conditions hold. (1) Commutative law for addition: For all elements u and v in V, u + v = v + u (2) Associative law for addition: For all elements u, v, w in V, u + (v + w) = (u + v) + w (3) Additive identity: The set V contains an additive identity element, denoted by 0, such that for any element v in V, v + 0 = v. (4) Additive inverses: For each element u in V, the equation v + x = 0 has a solution x in V, called an additive inverse of v. (5) Unitary law: For all elements v in V, lv = v (6) Distributive law I: For all real numbers a and all elements u, v in V, c(u + v) = cu + cv (7) Distributive law II: For all real numbers c, d and all elements v in V, (c+d)v = cv + dv (8) Associative law for multiplication: For all real numbers c, d and all elements v in V, c(dv) = (cd)v
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