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ilyas.h
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Homework Statement
Let V be a vector space over a field F and let L and M be two linear transformations from V to V.
Show that the subset W := {x in V : L(x) = M(x)} is a subspace of V .
The Attempt at a Solution
I presume it's a simple question, but it's one of those where you just don't know where to start. This is my attempt;
Subspace:
-0 vector exists
-closed under addition
-closed under scalar multiplicationLinear transformation:
-x,y in V: L(x+y) = L(x) + L(y)
-a in F, x in V: L(ax) = aL(x)
Proving 0 vector exists:
L & M are linear transformations so:
L(x+y) = L(x) + L(y)
but since L(x) = M(x):
L(x+y) = M(x) + M(y) = M(x+y)
∴ L(x+y) + (-M(x+y)) = 0_v
Also:
L(ax) = aL(x)
but since L(x) = M(x):
L(ax) = aM(x) = M(ax)
∴ L(ax) + (-M(ax)) = 0_v
Therefore, 0 vector exists in W.That's all I've got. Don't know how to prove closed under addition and closed under scalar multiplication.
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