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Question
I don't understand what has to be done for this problem. Ay help would be greatly appreciated.

(a) Show that for any choice of y_1, y_2, y_3, there is one and only one parabola y = ax^2+bx+c
passing by the points (1, y_1), (2, y_2) and (3, y_3).
(b) Show that the span of the vectors <1,4,9> , <1,2,3> and <1,1,1> is all of R^3.
(c) Explain briefly the link between (a) and (b).

Answer
a) Each of the points must satisfy the parabolic equation, so

(1, y1) -> a+b+c = y1
(2, y2) -> 4a+2b+c = y2
(3, y3) -> 9a+3b+c = y3

or in matrix form A・(a,b,c)^T = (y1,y2,y3)^T

( 1  1  1 )(a)      ( y1 )
( 4  2  1 )(b)  = ( y2 )
( 9  3  1 )(c)      ( y3 ).

This has a unique soution if the matrix, A, has a non-zero determinant.

detA = (1)(2-3)-(1)(4-9)+(1)(12-18) = -2 ≠ 0.

b) In order to span R^3, the 3-D vectors (1,4,9), (1,2,3) and (1,1,1) must be linearly indpendent, which means the matrix whose rows are made up of the vectors has to have a non-zero determinant

         | 1  4  9 |
det(B) =  | 1  2  3 | = -2 ≠ 0.
         | 1  1  1 |

c) The matrices A and B are both square and A^T = B which means that the linear indepence of the vectors shown by the non-zero determinant of B could have been deduced immediatey by the linear independence of the vectors in A.

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college mathematics, applied math, advanced calculus, complex analysis, linear and abstract algebra, probability theory, signal processing, undergraduate physics, physical oceanography

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