MATH307 December 2005
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Question 05 (a)
Let X be the subspace in three dimensional space containing all vectors perpendicular to
Let be the linear transformation defined by the first projective vector onto the plane and then projecting the resulting vector onto X.
(a) Show that the vectors and form a basis for .
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To prove that a set of vectors form a basis, the set of vectors have to:
(1) be linearly independent, and
(2) span contains all vectors perpendicular to
from the first row, we get from the third row, we get , which means
Since is the only solution, and are linearly independent.
(2) For arbitrary matrix that exists in X: ,
Since the dot product = 0, we have proven that all vectors in the span of X is perpendicular to the matrix