MATH221 April 2013
• Q1 • Q2 (a) • Q2 (b) • Q2 (c) • Q2 (d) • Q3 • Q4 • Q5 • Q6 • Q7 (a) • Q7 (b) • Q7 (c) • Q8 • Q9 • Q10 • Q11 • Q12 (a) • Q12 (b) • Q12 (c) •
Find a basis for the orthogonal complement of W.
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Are all three of the vectors linearly independent? If not, what is the dimension (and basis) of W. What will the dimension of the basis of be?
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We first note that there is a non-trivial linear relationship between the three vectors, namely
Thus, we get that has dimension 2 and thus is also 2 dimensional. Let be the matrix of basis vectors to ,
If is a basis vector to then orthogonality tells us that
Computing this we get the following relationships
Since the dimension (or rank) of V is two we will have two free parameters. In particular, this means that the basis is not unique (it never is). We choose and as free parameters to get
and finally that
Therefore the two orthogonal vectors are a basis for . Notice that the dimension of is two, which happens to be the same dimension as that of .
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