User:Eml5526.s11.team01.roark/Mtg17

[[media: Fe1.s11.mtg17.djvu| Mtg 17]]: Wed, 9 Feb 11 [[media: Fe1.s11.mtg17.djvu| Page 17-1]]

Question:

Linear independence of $$\displaystyle \left\{ {{x}^{j}},~j=0,1,2,\ldots \right\}$$

Answer: determinant $$\displaystyle \text{ }\!\!\mathbf \Gamma\!\!\text{ }\left( \left\{ {{x}^{j}} \right\} \right)\ne 0$$on $$\displaystyle \text{ }\!\!\Omega\!\!\text{ }=\left] a,b \right[$$ Any func $$\displaystyle {{x}^{k}}$$ (k fixed, e.g, k=0) cannot be expressed as a linear combination of$$\displaystyle \left\{ {{x}^{j}};j\ne k \right\}$$. In others words, $$\displaystyle \exists $$ no $$\displaystyle \left\{ {{c}_{1}},{{c}_{2}},{{c}_{3}},\ldots \right\}$$ such that $$\displaystyle {{x}^{0}}=1=\underset{\begin{matrix} j=1, \\ j\ne 0 \\ \end{matrix}}{\overset{\infty }{\mathop \sum }}\,{{c}_{j}}{{x}^{j}}$$ [[image: Eml5526.s11.roark.figure17-1.svg]]

End Question Section

HW 3.7:

Do HW3.1 with k=0

End HW 3.7

HW 3.8:

FB, P. 72, problem 3.1.

End HW 3.8

HW 3.9:

FB, P. 72, problem 3.3.

End HW 3.9

HW 3.10:

FB, P. 72, problem 3.4.

End HW 3.10

HW 3.11:

Do HW 2.9 with full Fourier basis $$\displaystyle \left\{ 1,cosx,sinx,cos2x,sin2x,... \right\}\to {{{b}'}_{j}}(0)\ne 0$$ For some j.

End HW 3.11

Note:

Discussed choices of basis functions, explained linear independence, Gram matrix.

End Note Section