MATH221 December 2007
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Find a basis for the null space of the matrix
Your answer will depend on t.
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To determine a basis for the null space of A, let and suppose Ax = 0. We want to solve this equation for x.
To do so, we append the zero vector to A
and row reduce A in the standard way. Leaving aside these details (exercise), the end result is
Therefore we deduce that x belongs to the null space of A if and only if
Case 1: . Then , so that
where is a free parameter. In this case we therefore conclude that is a basis for the null space of A.
Case 2: . Then there is no restriction on , and we have
where . In this case, we conclude that is a basis for the null space of A.
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