Page 351 - 35Linear Algebra

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(a) Write down a linear system of equations you could use to find the slope
m and constant term b.
(b) Arrange the unknowns (m, b) in a column vector X and write your
answer to (a) as a matrix equation

MX = V .

Be sure to give explicit expressions for the matrix M and column vector
V .
(c) For a generic data set, would you expect your system of equations to
have a solution? Briefly explain your answer.

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(d) Calculate M M and (M M) −1 (for the latter computation, state the
condition required for the inverse to exist).
(e) Compute the least squares solution for m and b.
(f) The least squares method determines a vector X that minimizes the
length of the vector V − MX. Draw a rough sketch of the three data
points in the (x, y)-plane as well as their least squares fit. Indicate how
the components of V − MX could be obtained from your picture.

Solutions

1. You can find the definitions for all these terms by consulting the index of
this book.

2. Both junctions give the same equation for the currents

I + J + 13 = 0 .

There are three voltage loops (one on the left, one on the right and one going
around the outside of the circuit). Respectively, they give the equations
60 − I − 80 − 3I = 0

80 + 2J − V + 3J = 0

60 − I + 2J − V + 3J − 3I = 0 . (F.1)

The above equations are easily solved (either using an augmented matrix
and row reducing, or by substitution). The result is I = −5 Amps, J = −8
Amps, V = 40 Volts.

3. (a) m.

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