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Written Solution to Questions #1,2,5,6,7 Written Solution to Questions #1,2,5,6,7 Written Solution to Questions #1,2,5,6,7 Written Solution to Questions #1,2,5,6,7 Written Solution to Questions #1,2,5,6,7
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Explanation & Answer
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Question 1:
We are given that
x1
x
2
U = x = : x1 + x2 + x3 + x4 = 0
x3
x
4
In U , we can re-write x1 = − x2 − x3 − x4 so that x can be expressed by
x1
x
x = 2
x3
x4
− x2 − x3 − x4
x2
x=
x3
x4
−1
−1
−1
1
0
0
x = x2
+x
+x
0 31 40
0
0
1
−1 −1 −1
0
0
1
Hence, the basis for U is BU = , , , and the dimension of U is 3 .
0 1 0
0 0 1
By definition of orthogonal component, we have
−1 1 0 0 y = 0
y U ⊥ −1 0 1 0 y = 0
−1 0 0 1 y = 0
y1
y
2
Now, if we let y = where y1 , y2 , y3 , y4 R then from the above matrix equations we have
y3
y4
− y1 + y2 = 0
− y1 + y3 = 0
− y + y = 0
1 4
y1 = y2 = y3 = y4
y1
y
2
which means that for y = U ⊥ , we have y1 = y2 = y3 = y4 .
y3
y4
Hence,
y1
y
2
U⊥ = y =
: y1 = y2 = y3 = y4
y3
y
4
1
1...