Question

Solution to 3. is there a linear transformation t : r 3 → r 2 such that t(1, −1, 1) = (1, 0) and t(1, 1, 1) = (0, 1)?

Answer 1

f(0) = 2, f(−1) = 1, f(1) = 3 and f(3) = −1.
Solution. By Lagrange Interpolation,
f(x) = 2
(x + 1)(x − 1)(x − 3)
(0 + 1)(0 − 1)(0 − 3)
+ 1
x(x − 1)(x − 3)
(−1)(−1 − 1)(−1 − 3)
+ 3
x(x + 1)(x − 3)
1(1 + 1)(1 − 3)
+ (−1)
x(x + 1)(x − 1)
3(3 + 1)(3 − 1)
=
2
3
(x + 1)(x − 1)(x − 3) −
1
8
x(x − 1)(x − 3)
−
3
4
x(x + 1)(x − 3) −
1
24
x(x + 1)(x − 1).