Classroom Assessment - Monday, 10/22

Let T: Rⁿ -> R^m be a linear transformation with standard matrix A.

Q1. T is one to one if and only if A has _______ pivot columns.

Q2. If T is onto, what is relation between m and n?

Q3. True or False? T is one to one if and only if for every x, there is a unique y with T(x) = y.

Answers:

Q1. The matrix A must have n pivot columns.

Reason: T is one to one iff the only solution to Ax = 0 is x = 0. This means "no free variables" in solution, which happens when there is a pivot in every column. Since the number of columns is n, then there are n pivot columns.

Q2. The number of columns is greater than or equal to the number of rows: n > m or n = m.

Reason: T is onto iff A reduces to a row echelon matrix with no zero row. This happens when there is a pivot in every row of A, so the number of pivots is m. There is at most one pivot in each column, so the number of pivot columns is m, and the total number of columns, which is n, must be greater or equal to m.

Q3. False. The statement about T is true for all linear transformations T.

Reason: One must read carefully what this statement says. It says that for any x, there is exactly one T(x). But this just means that the rule for T always produces a y = T(x) for all x. This just means that T is defined. It says nothing special about T. The definition for one-to-one says that for any y in the range of T, there is exactly one x. Not the same thing at all.