Let T be a linear transformation and let A denote its standa

Let T be a linear transformation and let A denote its standard matrix

Let T : R5 Rightarrow R4 be a linear transformation, and let A denote its standard matrix. If T is one-to-one, how many pivots must the RREF of A have? Is this possible? If T is onto, how many pivots must the RREF of A have? Is this possible?

Solution

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 5, then there are 5 pivot columns.

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 4. There is at most one pivot in each column, so the number of pivot columns is 4, and the total number of columns, which is 5, must be greater or equal to 4.