If vectors u, v, and w are linearly independent, will u + v, v + w, and u + w also be linearly independent? Justify your answer.
To determine if a set of vectors is linearly independent, it is sufficient to show that if a linear combination of the vectors is equal to zero, then each coefficient must be zero.
To determine if a set of vectors is linearly dependent, it is sufficient to show that a nontrivial solution to a linear combination of the vectors equals zero.
Row reducing the associated coefficient matrix gives a matrix of rank 2.
Since u, v, and w are linearly independent, then c1 + c3 = 0, c1 + c2 = 0, and c2 + c3 = 0.
Let the linear combination c1(u + v) + c2(v + w) + c3(u + w) = (c1 + c3)u + (c1 + c2)v + (c2 + c3)w = 0.
Therefore, the vectors u + v, v + w, and u + w are linearly dependent.
Construct the proof by choosing the appropriate statements from the list and putting them in the correct order.
(b) If vectors u, v, and w are linearly independent, will u - v, v - w, and u - w also be linearly independent? Justify your answer.
To determine if a set of vectors is linearly independent, it is sufficient to show that if a linear combination of the vectors is equal to zero, then each coefficient must be zero.
To determine if a set of vectors is linearly dependent, it is sufficient to show that a nontrivial solution to a linear combination of the vectors equals zero.
Since u, v, and w are linearly independent, then c1 + c3 = 0, -c1 + c2 = 0, and -c2 - c3 = 0.
Row reducing the associated coefficient matrix gives a matrix of rank 3.
Let the linear combination c1(u - v) + c2(v - w) + c3(u - w) = (c1 + c3)u + (-c1 + c2)v + (-c2 - c3)w = 0.
Therefore, the vectors u - v, v - w, and u - w are linearly independent.
Construct the proof by choosing the appropriate statements from the list and putting them in the correct order.