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# Linear Algebra: Basis and Dimension

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.

See attached file for full problem description.

#### Solution Preview

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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