A cart moves along a straight rail

PHYSICS · Multi-part · Fermi University Physics · Scientific Practice & Experimental Methods · Mathematical Methods for Physics · Data analysis and model fitting in the laboratory

Conditions of Validity, Ch Inner products, orthogonality and the adjoint

Problem

A cart moves along a straight rail. Its position

y

(in

\mathrm{m}

) is recorded at times

x

(in

\mathrm{s}

): at

x=1,\ 2,\ 3,\ 4\ \mathrm{s}

the positions are

y=2.1,\ 3.9,\ 6.2,\ 7.8\ \mathrm{m}

. Model the motion as

y = a + bx

, with

a

\mathrm{m}

and

b

\mathrm{m\,s^{-1}}

, and treat

\mathbb{R}^4

with the ordinary dot product as the inner product space. ((a)) Writing the model as

\mathsf{A}\mathbf{p} = \mathbf{b}

with

\mathbf{p} = (a,b)^{\mathrm{T}}

and columns

(1,1,1,1)^{\mathrm{T}}

and

(1,2,3,4)^{\mathrm{T}}

, use the projection theorem to derive the normal equations

\mathsf{A}^{\dagger}\mathsf{A}\mathbf{p} = \mathsf{A}^{\dagger}\mathbf{b}

in the explicit form

n a + b\sum x_i = \sum y_i, \qquad a\sum x_i + b\sum x_i^2 = \sum x_i y_i.

((b)) Solve the normal equations for

a

and

b

using the recorded data. ((c)) Let

\mathbf{u} = (x_i - \bar{x})

and

\mathbf{v} = (y_i - \bar{y})

be the centred data vectors. State the Cauchy-Schwarz inequality for

\mathbf{u}, \mathbf{v}

and use it to show that the correlation coefficient

r = \langle\mathbf{u},\mathbf{v}\rangle / (\|\mathbf{u}\|\,\|\mathbf{v}\|)

satisfies

|r| \le 1

. Compute

r

for the data and check the bound holds.

Browse the problem library