MATH307 December 2008
• Q1 • Q2 • Q3 • Q4 • Q5 • Q6 (a) • Q6 (b) • Q7 • Q8 • Q9 • Q10 •
Determine the rank of A and find a basis of the left null space N(AT) of A.
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Usually the quickest way to find the rank of a matrix A is to use row-reduction. But hold your horses: Since we need to work with AT anyway we can get the rank of A by considering AT only.
To find the left null space N(AT) row-reduce AT.
Since the rank of A equals the rank of AT, row-reducing AT will also reveal the rank of A.
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The rank can be found from the rref of A or of however is easier to find for this matrix, and necessary to calculate N(AT). So we start by calculating .
Now calculating is quite easy because all of the rows are multiples of the first row and hence linearly dependent. See that . Hence
The rank of A is equal to the number of leading zeroes in rref(A) so the rank of A is 1.
Now to find the nullspace of we start by writing the matrix as an equation equal to zero.
Now since there are two independent variables, N(A) will have two vectors. We can find them by setting up a vector with the equation we found above.
And therefore we find that
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