Math Solutions

(Solution Library) Student group membership. Let G ∈ R^m * n represent a contingency matrix of m students who are members of n groups: G_i j= \begincases1

Question: Student group membership. Let \(G \in \mathbf{R}^{m \times n}\) represent a contingency matrix of \(m\) students who are members of \(n\) groups:

\[G_{i j}= \begin{cases}1 & \text { student } i \text { is in group } j \\ 0 & \text { student } i \text { is not in group } j\end{cases}\]

(A student can be in any number of the groups.)

  1. What is the meaning of the 3rd column of \(G\) ?
  2. What is the meaning of the 15 th row of \(G\) ?
  3. Give a simple formula (using matrices, vectors, etc.) for the \(n\) -vector \(M\), where \(M_{i}\) is the total membership (i.e., number of students) in group i .
  4. Interpret \(\left(G G^{T}\right)_{i j}\) in simple English.
  5. Interpret \(\left(G^{T} G\right)_{i j}\) in simple English.

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