Linear Algebra with Applications (3rd Edition)

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THE JOHNS HOPKINS UNIVERSITY Faculty of Arts and Sciences FIRST TEST - SPRING SESSION 2005 110.201 - LINEAR ALGEBRA. Examiner: Professor C. Consani Duration: 50 minutes, March 9, 2005 No calculators allowed. Total Marks = 100 1. Let A be a ( m × n )-matrix of rank r . Suppose AX = b has no solution for some right sides b and infinitely many solutions for some other right sides b . (a) [ 5 marks] Decide whether the nullspace of A contains only the zero vector and why. Sol. If the nullspace of A ( N ( A )) contains only the zero vector, then dim N ( A ) = n - r = 0, i.e. n = r (with n m ). But then, the system could not have infinitely many solutions for some b . (b) [ 5 marks] Decide whether the column space of A is all of R m and why. Sol. If the column space of A were R m , then the system would have always a solution, but this is in contradiction with the hypothesis that for some b the system has no solution. (c) [ 5 marks] For this matrix A find all true relations between the numbers r , m and n . Sol. It is always true that r m and r n . Moreover, under the given hypothesis, we must have r < m (otherwise there would not be right sides b such that AX = b has no solution) and r < n (otherwise there would not be any b such that AX = b has infinitely many solutions). (d) [ 5 marks] Can there be a right side b for which AX = b has exactly one solution? Why or why not? Sol. No, because this condition would require n = r . 2. (a) [ 5 marks] Are the vectors v 1 = - 2 - 1 3 4 and v 2 = - 8 2 - 2 1 linearly independent? Are these vectors perpendicular to each other? Explain your answers.
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