Prove that C = R2 as vector spaces over R and use this to show that.pdf
0 likes8 views
The document demonstrates that the complex numbers (C) can be treated as a vector space isomorphic to R^2, thereby establishing that the dimension of C is 2. It constructs a linear map from C to R^2 by mapping a complex number z = x + iy to the ordered pair (x, y). The proof shows the isomorphism is both one-to-one and onto, affirming the relationship and the significance of graphing complex numbers in the xy-plane.
1 of 1
Download to read offline
Prove that C = R2 as vector spaces over R and use this to show that.pdf
- 1. Prove that C ?= R2 as vector spaces over R and use this to show that dim(C) = 2. (Hint: Construct a map from C to R2 and prove that this map is linear and an isomorphism.) Note: This isomorphism is why we can graph complex numbers using the standard xy-plane. Solution Take any element z in C z = x+iy Consider the mapping from C to R2 as z =x+iy is mapped onto (x,y) in R2. Then this is obviously one to one and onto mapping. To prove that dim C =2 we have 1 and i as scalars such that z = 1x+iy Hence dim C =2 Z1+z2 = x1+iy1+x2+iy2= (x1+x2) +i(y1+y2) thus isomorphism is preserved This isomorphism is why we can graph complex numbers using Argand diagram in xy plane.