Practice Questions Lecture # 7 and 8 Question # Determine whether the following system has a trivial solution or non-trivial solution: x x + x x x x x The coefficient matrix is / R, R R R+ R The corresponding system of equations will be x x + x x x x By backward substitution, we get x, x, x. Thus, the given system has only the trivial solution. Question # Solve the following system using the reduced echelon form: x + x + 6x x 5x x 7x Given system of equation could be translated into matrix form as
- -5-7 add times the st row to the nd row 8 7-7 Multiply the nd row by /8 7 8-7 add - times the nd row to the rd row 7 8-77 8 Multiply the rd row by -8/77
7 8 add -7/8 times the rd row to the nd row add -6 times the rd row to the st row add - times the nd row to the st row Question # Check whether { v, v, v } is linearly dependent or not? where v, v and v The set S {v, v, v } of vectors in R is linearly independent if the only solution of c v + c v + c v is c, c, c Otherwise (i.e., if a solution with at least some nonzero values exists), S is linearly dependent.
With our vectors v, v, v, becomes: - c + c + c - Rearranging the left hand side yields - c + c + c c + c - c c + c + c The matrix equation above is equivalent to the following homogeneous system of equations - c + c + c c + c - c c + c + c We now transform the coefficient matrix of the homogeneous system above to the reduced row echelon form to determine whether the system has the trivial solution only (meaning that S is linearly independent), or the trivial solution as well as nontrivial ones (S is linearly dependent). - - can be transformed by a sequence of elementary row operations to the matrix The reduced row echelon form of the coefficient matrix of the homogeneous system is
which corresponds to the system c c c Since each column contains a leading entry (highlighted in yellow), then the system has only the trivial solution, so that the only solution is c, c, c. Therefore the set S {v, v, v } is linearly independent. Question # Determine, without solving, whether the following set of vectors is linearly independent or 7 dependent. S,,, 6 Using the same rule in previous question, with our vectors v, v, v, v c + c + c 6 7 + c Rearranging the left hand side yields c + c + c +7 c c + c +6 c + c The matrix equation above is equivalent to the following homogeneous system of equations c + c + c +7 c
c + c +6 c + c We now transform the coefficient matrix of the homogeneous system above to the reduced row echelon form 7 6 can be transformed by a sequence of elementary row operations to the matrix -9 The reduced row echelon form of the coefficient matrix of the homogeneous system is -9 which corresponds to the system c + c +(-9/) c c + c +(/) c The leading entries have been highlighted in yellow. Since some columns do not contain leading entries, then the system has nontrivial solutions, so that some of the values c, c, c, c solving the system of equation may be nonzero. Therefore the set S {v, v, v, v } is linearly dependent.
Question # 5 Show that the columns of A are linearly independent. Vectors in R is linearly independent if the only solution of c v + c v is c, c c + c - Rearranging the left hand side yields c + c c - c The matrix equation above is equivalent to the following homogeneous system of equations c + c c - c We now transform the coefficient matrix of the homogeneous system above to the reduced row echelon form - can be transformed by a sequence of elementary row operations to the matrix
The reduced row echelon form of the coefficient matrix of the homogeneous is which corresponds to the system c c Since each column contains a leading entry (highlighted in yellow), then the system has only the trivial solution, so that the only solution is c, c. Therefore columns of A are linearly independent.