Question

Consider V = ℝ3, F= ℝ Let U be the XY plane and W be the Z−axis. Show that V = U ⊕ W.

Answer 1

Answer:

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Answer 2

V = U ⊕ W is proved where V = ℝ³, F= ℝ U be the XY plane and W be the Z−axis

Given : V = ℝ³, F = ℝ , U be the XY plane and W be the Z−axis

To find : To prove V = U ⊕ W.

Solution :

Here it is given that V = ℝ³, F = ℝ , U be the XY plane and W be the Z−axis

Thus we have

V = {(x, y, z) : x, y, z ∈ ℝ }

U = The XY plane

= {(x, y, 0) : x, y ∈ ℝ }

W = The Z−axis

= {(0,0, z) : z ∈ ℝ }

Clearly U ⊆ V and W ⊆ V

Thus we get U + W ⊆ V - - - - - (1)

Let (x, y, z) ∈ V

Then (x, y, z) = (x, y, 0) + (0,0,z)

Thus we have V ⊆ U + W - - - - - (2)

So we get from Equation 1 and Equation 2

V = U + W

Next suppose that

(x, y, z) ∈ U ∩ W

Now

(x, y, z) ∈ U implies z = 0

(x, y, z) ∈ W implies x = 0 and y = 0

So (x, y, z) ∈ U ∩ W implies x = 0 , y = 0 , z = 0

So U ∩ W = Φ

Thus we can conclude that V = U ⊕ W.

Hence the proof follows

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