Question 4.E.3.12: Suppose that S = {u1, u2, . . . , un} is a set of vectors fr......

Matrix Analysis and Applied Linear Algebra [1113429]

Suppose that $\mathcal{S} = \{u_{1}, u_{2}, . . . , u_{n}\}[$ is a set of vectors from $ℜ^m$ . Prove that $\mathcal{S}$ is linearly independent if and only if the set

$\mathcal{S}^{′}=\left\{u_{1},\sum\limits_{i=1}^{2}u_{i}, \sum\limits_{i=1}^{3}u_{i}, . . . ,\sum\limits_{i=1}^{n}u_{i}\right\}$

is linearly independent.

Step-by-Step

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If $A_{m×n}$ is the matrix containing the $u_{i}’s$ as columns, and if

$Q_{n×n} =\begin{pmatrix}1 &1& · · ·& 1\\0 &1& · · ·& 1\\\vdots&\vdots&\ddots&\vdots\\0 &0& · · · &1\end{pmatrix},$

then the columns of B = AQ are the vectors in $\mathcal{S}^{′}$ . Clearly, Q is nonsingular so that A $\overset{col}{\sim}$ B, and thus rank (A) = rank (B). The desired result now follows from (4.3.3).

rank (A) = n. (4.3.3)

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