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Name: 1. Let $\mathbb{C} = \{a+bi : a, b \in \mathbb{R}\}$. Here $i^2 = -1$. Given $a+bi, c+di \in \mathbb{C}$ and a real number $r$, we define addition by $(a+bi)+(c+di) = (a+c)+(b+d)i$, and scalar multiplication by $r(a+bi) = ra + (rb)i$. Show that $\mathbb{C}$ is a vector space.

Transcript

00:01 Hi, in this question given that c is the complex number system and also given that define the addition of c as a plus b i plus c plus d i equals a plus c plus b into i and also scalar multiplication of complex number is alpha into a plus b i which is equal to alpha a plus alpha b into i here we need to prove that complex number system is a vector space we know that vector space must satisfy the eight axioms so here for all x y z in complex number system and alpha comma beta in real number system so first property is associativity, that is x plus y plus z equals x plus y plus z.

01:17 So here we have to consider a plus b i plus that is a1 plus b1 i.

01:29 Here a2 plus b2i plus a3 i which can be written as a1 1 plus a 2 plus a 3 plus i enter b1 plus b2 plus b 3 by using the given addition property so it satisfies the associative property next property is x plus y equals y plus x that is it must satisfy the commutative property so here we have to consider a plus i b plus c plus i d which can be written as a plus c plus i into b plus d since a b c d are reals and they are commutative so we can write it as c plus a plus a plus i into d plus b which is equal to c plus id plus a plus a plus i b hence it satisfies the second axiom that is commutative property.

02:49 Next third property, that is, there exist zero in c, such that 0 plus x equals x.

03:04 Here we have to consider 0 plus 0 i plus a plus i b, which is equal to 0 plus a plus i b, which is equal to 0 plus a plus i into 0 plus b which is equal to a plus i b so it is also satisfies third property that is identity for fourth property for each v in c there exists minus v in c such that minus v plus v equal to so here we have to consider minus of a plus i b equals minus a plus i into minus b so here we can write it as minus of a plus i b plus a plus i b equals minus a plus a plus i b plus i which is equal to zero plus i so which is the zero vector so hence it is prove the fourth property next we need to prove the fifth property that is alpha of x plus y equal alpha x plus alpha y we can use the scalar that is alpha into a plus i b plus c plus i d which is equal to alpha into a plus c plus i into b plus d by using the scale or multiplication, then we get alpha into a plus c plus i alpha into b plus d.

04:54 Since alpha a, b, c, d or reals and multiplication distributive over addition.

05:01 So it satisfies the fifth property...

Question

Name: 1. Let $\mathbb{C} = \{a+bi : a, b \in \mathbb{R}\}$. Here $i^2 = -1$. Given $a+bi, c+di \in \mathbb{C}$ and a real number $r$, we define addition by $(a+bi)+(c+di) = (a+c)+(b+d)i$, and scalar multiplication by $r(a+bi) = ra + (rb)i$. Show that $\mathbb{C}$ is a vector space.

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