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Ted123
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Homework Statement
Suppose [itex]\phi : V \to W[/itex] is a linear transformation over a field [itex]\mathbb{K}[/itex] .
(a) (i) Define the kernel of [itex]\phi, \; Ker(\phi)[/itex]. [2 marks]
(ii) Show that [itex]\phi[/itex] is 1-1 if and only if [itex]Ker ( \phi )=\{\bf{0}\}[/itex]. [4 marks]
(b) Suppose X is a subspace of W. Define the set [itex]U_X[/itex] by
[itex]U_X = \{ v\in V : \phi (v) \in X \}[/itex] .
Show that [itex]U_X[/itex] is a subspace containing [itex]Ker ( \phi )[/itex]. [7 marks]
(c) Suppose that Y is another subspace of W. Show that [itex]U_X \cap U_Y = U_{X \cap Y}[/itex]. [4 marks]
(d) What does it mean to say that W is the direct sum of two subspaces X and Y (written [itex]W=X \oplus Y[/itex] )? [2 marks]
(e) Show that if [itex]W=X \oplus Y[/itex] then [itex]U_X \cap U_Y = Ker(\phi)[/itex]. [6 marks]
Total [25 marks]
The Attempt at a Solution
Done (a)
For (b), [itex]\phi(\bf{0})=\bf{0} \in X[/itex] since X is a subspace, so [itex]\bf{0}\in U_X[/itex] and [itex]U_X[/itex] is non-empty.
Suppose [itex]u,v\in U_X[/itex] .
Then [itex]\phi(u+v) = \phi(u) + \phi(v) \in X[/itex] since X is a subspace, so [itex]U_X[/itex] is closed under vector addition.
Suppose [itex]\alpha \in \mathbb{K}[/itex] and [itex]v\in U_X[/itex].
Then [itex]\phi(\alpha v) = \alpha \phi(v) \in X[/itex] since X is a subspace.
Hence [itex]U_X[/itex] is a subspace containing [itex]\bf{0}[/itex] and hence [itex]Ker( \phi )[/itex] .
For (c) is this alright?
[itex]U_X \cap U_Y = \{ v\in V : \phi(v) \in X \} \cap \{ v\in V : \phi(v) \in Y \}[/itex]
[itex]= \{ v\in V : \phi(v) \in X\;\text{and}\;\phi(v)\in Y \}[/itex]
[itex]= \{ v\in V : \phi(v) \in X \cap Y \} = U_{X\cap Y}[/itex]
For (d), the direct sum [itex]W=X \oplus Y[/itex] means [itex]W=X+Y[/itex] and [itex]X\cap Y = \{ \bf {0}\}[/itex]
For (e) is this alright?
If [itex]W=X \oplus Y[/itex] then [itex]X\cap Y = \{ \bf {0}\}[/itex] . So
[itex]U_X \cap U_Y = U_{X \cap Y}[/itex] (by part (c))
[itex]= U_{\{\bf{0}\}} = \{v\in V : \phi(v)\in \{\bf{0}\} \} = \{v\in V : \phi(v) = \bf{0} \} = Ker( \phi )[/itex]
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