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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).

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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