Suppose t1 : r 3 ? r 2 is given by t1
((x1))
((x2))
((x3)) =
((2x1 + 3x2))
+ x3 ))
and t2 : r 2 ? r 3 is given by t2
((w1))
((w2)) =
((w1 + w2))
((2w1 ? w2))
((w1 + 3w2)) .
(a) find the standard matrix representing the linear transformation t2 ? t1. (b) calculate (t2 * t1)(x)
where x =
((1 ))
((2))
((3))
using the matrix you found, and then verify the result using the definitions of t1 and t2.
(2) if asked to determine whether a 3 × 3 matrix has an inverse, and calculate the inverse if it exists, recall that the usual algorithm is to “augment the matrix with the identity matrix and use gaussian elimination”. (a) why do we “augment with the identity matrix”? (hint: consider what augmenting with each column represents.) (b) if the matrix is invertible, what do we get on the left side after completing gaussian elimination, and why? (c) if the matrix is invertible, why does the inverse appear on the right side? (hint: consider elementary matrices, or the row operations they represent.)