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The set C of all complex numbers, with the usual operations of addition and real scalar multiplication, forms a vector space. Elements of C can be written in the form æ + iy, where i is the imaginary unit. 1. Show that closure is satisfied for C under addition and scalar multiplication. What is the zero "vector" in this vector space? 2. Come up with two different subspaces of C other than (0} and C itself. Explain how they satisfy the definition of a subspace. 3. R is closed under multiplication; in other words, the product of two elements Ris also in R. Can we say the same for C? Why or why not?

The set C of all complex numbers, with the usual operations of addition and real scalar multiplication, forms a vector space. Elements of C can be written in the form æ + iy, where i is the imaginary unit. 1. Show that closure is satisfied for C under addition and scalar multiplication. What is the zero "vector" in this vector space? 2. Come up with two different subspaces of C other than (0} and C itself. Explain how they satisfy the definition of a subspace. 3. R is closed under multiplication; in other words, the product of two elements Ris also in R. Can we say the same for C? Why or why not?

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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The set C of all complex numbers, with the usual operations of addition and real scalar multiplication, forms a vector space. Elements of C can be written in the form x + iy, where i is the imaginary unit. 1. Show that closure is satisfied for C under addition and scalar multiplication. What is the zero "vector" in this vector space? 2. Come up with two different subspaces of C other than {0} and C itself. Explain how they satisfy the definition of a subspace. 3. R is closed under multiplication; in other words, the product of two elements in R is also in R. Can we say the same for C? Why or why not?
Transcribed Image Text:The set C of all complex numbers, with the usual operations of addition and real scalar multiplication, forms a vector space. Elements of C can be written in the form x + iy, where i is the imaginary unit. 1. Show that closure is satisfied for C under addition and scalar multiplication. What is the zero "vector" in this vector space? 2. Come up with two different subspaces of C other than {0} and C itself. Explain how they satisfy the definition of a subspace. 3. R is closed under multiplication; in other words, the product of two elements in R is also in R. Can we say the same for C? Why or why not?
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