Consider Example 12 in section 6.2 (page 311). Study each space closely and atte

Question

Consider Example 12 in section 6.2 (page 311). Study each space closely and attempt to understand how each is "the same" as the other. What mapping can you define to go from one to another? Is this mapping unique? Things that may seem more obvious in one space would have an analog in its isomorphic spaces, yet these concepts may not seem so obvious. Try to make the complete mental translation from one space to another. Conversely, can you now explain with great detail why the real line R1 and the Cartesian plane R2 are "not the same"? How would you explain this to a mathematician?

Explanation / Answer

for two spaces to be isomorphic

there should exist a one to one onto function such that

f(x+y)=f(x)+f(y)

and f(xy)=f(x)f(y)

if function is not one one then it vector spaces are called homomorphic

given spaces are all

isomorphic

all of them have 4 standard basis

and hence we can define one one onto functions on them

for

eg

standard basis for a

is

(0,0,0,1),(0,0,1,1),(0,1,0,0),(1,0,0,0)

for b)

[0,0,0,1]t,[0,0,1,0]t,[0,1,0,0]t[1,0,0,0]t

here t denotes transpose

for c)

we have four standard matrix as bases

1_0

0_0

,

0_1

0_0

,

0_0

1_0

,

0_0

0_1

for d)1,x,x^2,x^3

for e)(1,0,0,0,0),(0,1,0,0,0),(0,0,1,0,0),(0,0,0,1,0)

now choose any

two

let a and d

since both contains equal no of basis

define a one one onto function from a to b

now let f

is that function

now define

f as

f(x+y)=f(x)+f(y)

f(xy)=f(x)f(y)

now it is clearly seen that

both a nad b

have same

structure

i.e

for

any alement

in a)

we can write it as

a(0,0,0,1)+b(0,0,1,0)+c(0,1,0,0)+d(1,0,0,0)

and corresponding to this there exists a element in b

which can also be written in the same way

hence same structure

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