Question 4.E.3.3: Suppose that in a population of a million children the heigh......

Matrix Analysis and Applied Linear Algebra [1113429]

Suppose that in a population of a million children the height of each one is measured at ages 1 year, 2 years, and 3 years, and accumulate this data in a matrix

$\begin{matrix}\\\\\#1\\\#2\\\vdots\\\#i\\\vdots\end{matrix}\begin{matrix}\begin{array}{c c c c }1 yr&2 yr&3 yr\end{array}\\\begin{pmatrix}h_{11} &h_{12} &h_{13}\\h_{21}& h_{22}& h_{23}\\\vdots&\vdots&\vdots\\h_{i1}& h_{i2} &h_{i3}\\\vdots&\vdots&\vdots \end{pmatrix}\end{matrix} = H.$

Explain why there are at most three “independent children” in the sense that the heights of all the other children must be a combination of these “independent” ones.

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rank (H) ≤ 3, and according to (4.3.11), rank (H) is the maximal number of independent rows in H.

Any maximal independent subset of rows from A contains exactly r rows. (4.3.11)

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