Linear Algebra

106 Chapter Two. Vector SpacesRestated, independence is preserved by subset and dependence is preservedby superset.Those are two of the four possible cases of interaction that we can consider.The third case, whether linear dependence is preserved by the subset operation,is covered by Example 1.13, which gives a linearly dependent set S with a subsetS 1 that is linearly dependent and another subset S 2 that is linearly independent.That leaves one case, whether linear independence is preserved by superset.The next example shows what can happen.1.15 Example In each of these three paragraphs the subset S is linearlyindependent.For the set⎛ ⎞S = { ⎝ 1 0⎠}0the span [S] is the x axis. Here are two supersets of S, one linearly dependentand the other linearly independent.⎛dependent: { ⎝ 1 ⎞ ⎛0⎠ , ⎝ −3⎞⎛0 ⎠} independent: { ⎝ 1 ⎞ ⎛0⎠ , ⎝ 0 ⎞1⎠}0 00 0Checking the dependence or independence of these sets is easy.For⎛ ⎞ ⎛ ⎞S = { ⎝ 1 0⎠ , ⎝ 0 1⎠}0 0the span [S] is the xy plane. These are two supersets.⎛dependent: { ⎝ 1 ⎞ ⎛0⎠ , ⎝ 0 ⎞ ⎛1⎠ , ⎝ 3⎞⎛−2⎠} independent: { ⎝ 1 ⎞ ⎛0⎠ , ⎝ 0 ⎞ ⎛1⎠ , ⎝ 0 ⎞0⎠}0 0 00 0 1If⎛⎞⎛⎞⎛⎞S = { ⎝ 1 0⎠ , ⎝ 0 1⎠ , ⎝ 0 0⎠}0 0 1then [S] = R 3 . A linearly dependent superset is⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛⎞dependent: { ⎝ 1 0⎠ , ⎝ 0 10 0⎠ ,⎝ 0 01⎠ ,⎝ 2 −1⎠}3but there are no linearly independent supersets of S. The reason is that for anyvector that we would add to make a superset, the linear dependence equation⎛⎝ x ⎞ ⎛y⎠ = c 1⎝ 1 ⎞ ⎛0⎠ + c 2⎝ 0 ⎞ ⎛1⎠ + c 3⎝ 0 ⎞0⎠z 0 0 1has a solution c 1 = x, c 2 = y, and c 3 = z.