Statistics Solutions

[Solved] Linear independence of stacked vectors. Consider the stacked vectors

Question: Linear independence of stacked vectors. Consider the stacked vectors

\[c_{1}=\left[\begin{array}{l} a_{1} \\ b_{1} \end{array}\right], \ldots, c_{k}=\left[\begin{array}{l} a_{k} \\ b_{k} \end{array}\right]\]

where \(a_{1}, \ldots, a_{k}\) are \(n\) -vectors and \(b_{1}, \ldots, b_{k}\) are \(m\) -vectors.

  1. Suppose \(a_{1}, \ldots, a_{k}\) are linearly independent. (We make no assumptions about the vectors \(\left.b_{1}, \ldots, b_{k} .\right)\) Can we conclude that the stacked vectors \(c_{1}, \ldots, c_{k}\) are linearly independent?
  2. Now suppose that \(a_{1}, \ldots, a_{k}\) are linearly dependent. (Again, with no assumptions about \(\left.b_{1}, \ldots, b_{k} .\right)\) Can we conclude that the stacked vectors \(c_{1}, \ldots, c_{k}\) are linearly dependent?

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