Linear Algebra: Basis and Dimension
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10. Find the coordinate vector of p relative to the basis S = {p1, p2, p3}.
(a) p = 4 - 3x + x2; p1 = 1, p2 = x, p3 = x2
(b) p = 2 - x + x2; p1 = 1 + x, p2 = 1 + x2, p3 = x + x2
22. Find the standard basis vectors that can be added to the set {v1, v2} to produce a basis for R4.
v1 = (1, -4, 2, -3), v2 = (-3, 8, -4, 6)
30. Prove: Any subspace of a finite-dimensional vector space is finite-dimensional.
Hint: You can do a proof by contradiction by using 2 or 3 Theorems from section.
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10. Find the coordinate vector of p relative to the basis S = {p1, p2, p3}.
The coordinates of p relative to the basis S are the numbers c1, c2, c3 such that p = c1p1 + c2p2 + c3p3. The coordinate vector of p would be the vector {c1, c2, c3}. (Note how this relates to theorem 5.4.1 - there is only one coordinate vector for each p.)
(a) p = 4 - 3x + x2; p1 = 1, p2 = x, p3 = x2
p = c1(p1) + c2(p2) + c3(p3)
p = c1(1) + c2(x) + c3(x2)
c1 = 4, c2 = -3, c3 = 1
(b) p = 2 - x + x2; p1 = 1 + x, p2 = 1 + x2, p3 = x + x2
p = c1(p1) + c2(p2) + c3(p3)
p = c1(1 + x) + c2(1 + x2) + c3(x + x2)
2 - x + x2 = c1(1 + x) + c2(1 + x2) + c3(x + x2)
2 - x + x2 = c1 + c1x + c2 + c2x2 + c3x + c3x2
2 - x + x2 = (c1 + c2) + (c1+ c3)x + (c2 + c3)x2
2 = c1 + c2
-1 = c1+ c3
1 = c2 + c3
Solve this system ...
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