Homework Assignment 1
1. For each of the following equations determine whether it is linear in its variables or not. Explain your decision.
2. Solve each of the following systems and comment on geometric interpretation of its solutions.
a) b) c)
3. Solve the following linear systems by Gauss-Jordan method:
4. Each of the following matrices is an augmented matrix of some linear system. In each case, determine :
the ranks of the augmented and the coefficient matrices
if the system is consistent (explain your decision),
and what is the number of solutions if it is consistent.
5. Find the values of the parameters a and b such that the following linear system a) will be inconsistent; b) will have infinitely many solutions; c) will have exactly one solution.
6. Find all the values of a, for which the following homogeneous system has a non-trivial solution. Find the general solution in this case.
7. Propose an example of a REF of the augmented matrix of a system of 4 equations in 4 variables that (a) has a unique solution
(b) has infinitely many solutions
(c) is inconsistent.
8. For each of the following statements determine whether it is true (T) or false (F). Substantiate your decision. (a) A linear system is inconsistent if the number of its equations exceeds the number of the variables. (b) If the rank of the coefficient matrix of a system is equal to 5, then the rank of the augmented matrix cannot be equal to 7. (c) If the augmented matrix of a system in 6 variables has 4 pivot columns, then the system has infinitely many solutions.
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