- #1

Eclair_de_XII

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- 91

- Homework Statement
- Let ##X## be a vector space and ##Y## a subspace with ##\dim Y = \dim X##. Prove that ##Y=X##.

- Relevant Equations
- Quotient space of X mod Y:

##X/Y=\{x_1,x_2\in X:\,x_1-x_2\in Y\}##

Equivalence class of x w.r.t. quotient space:

##\{x\}_Y=\{x_0\in X:\,x_0-x\in Y\}\subset X/Y##

Theorem:

##\dim Y + \dim {X/Y} = \dim X##

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Assume that ##X/Y## is defined. Since ##\dim Y = \dim X##, it follows that ##\dim {X/Y}=0## and that ##X/Y=\{0\}##.

Suppose that ##Y## is a proper subspace of ##X##. Then there is an ##x\in X## such that ##x\notin Y##.

Let us consider the equivalence class:

##\{x\}_Y=\{x_0\in X:\,x_0-x\in Y\}##

This is a subset of ##X/Y=\{0\}##, which means that because ##\{x\}_Y## is a vector space, ##\{0\}\subset \{x\}_Y##. Hence, ##\{x\}_Y=\{0\}##.

This implies that ##0-x\in Y## and that ##x\in Y##, since ##Y## is closed under scalar multiplication. This contradicts the fact that ##x\notin Y##.

Hence, there cannot be an ##x\in X## that is not in ##Y##. Thus, ##X=Y##.

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I am a bit worried about the existence of ##X/Y##. Is it defined for every vector space? I am also worried that I am misinterpreting the meaning of quotient spaces and equivalence classes.

Assume that ##X/Y## is defined. Since ##\dim Y = \dim X##, it follows that ##\dim {X/Y}=0## and that ##X/Y=\{0\}##.

Suppose that ##Y## is a proper subspace of ##X##. Then there is an ##x\in X## such that ##x\notin Y##.

Let us consider the equivalence class:

##\{x\}_Y=\{x_0\in X:\,x_0-x\in Y\}##

This is a subset of ##X/Y=\{0\}##, which means that because ##\{x\}_Y## is a vector space, ##\{0\}\subset \{x\}_Y##. Hence, ##\{x\}_Y=\{0\}##.

This implies that ##0-x\in Y## and that ##x\in Y##, since ##Y## is closed under scalar multiplication. This contradicts the fact that ##x\notin Y##.

Hence, there cannot be an ##x\in X## that is not in ##Y##. Thus, ##X=Y##.

%%%

I am a bit worried about the existence of ##X/Y##. Is it defined for every vector space? I am also worried that I am misinterpreting the meaning of quotient spaces and equivalence classes.