Matrix Transform (Post Italicizing N Sequence) For this activity, think of each of the matrices listed below as a transformation T(x) = Ax = b where T:R" R". 1 00 1 3 12 20 0 0-1 0 3 [23] 0 0 2 For EACH matrix given, answer the following questions. (Your group will be assigned 1-3 matrices to have the primary responsibility for presenting, but try to do as many as possible in the time available.) 1. For your matrix, rewrite T:R" →R" with correct numbers for m and n filled in. 2. Find some way to explain in words and/or graphically what this transformation does in taking vectors from R" to R". (It may help to consider what happens to the standard basis vectors e₁, etc.) 3. Does the matrix/transformation have an inverse? i.e. Is there a way to get back from R" to R with every vector being sent back to the vector that it came from? 772 • If possible, find the matrix that does this. This is called the inverse matrix and is labeled, A¯\ . If it does not seem possible to reverse the process, what goes wrong? What causes the transformation to not be invertible? Now consider your answers to the above questions for all the matrices. Try other matrices also if it helps you answer the following question. 4. Explain as many strategies as you can think of for determining whether a given matrix will be the matrix of an invertible transformation. (In other words, describe in what cases a matrix will be invertible and in what cases it will not be invertible.) There are two main things that can go wrong with transformations to cause them to not be invertible: A. The transformation is not 1-1 (read: one to one). This means that for your matrix/transformation, you can find two different input vectors that will give the same output vector. In other words, the inverse would not be well defined because a vector would need to "come back to" more than one vector. B. The transformation is not onto. This means that for your matrix/transformation, there are some output vectors that cannot be reached. In other words, the inverse would not be well defined because some vectors would have NO vector to "come back to", since the origin transformation did not map to that vector. 1. For each of the matrices on the previous page tell whether or not the transformation is 1-1. • • If the matrix is NOT 1-1 show this by finding two vectors in R™ that get mapped to the same vector in R". If the matrix IS 1-1 explain why this is not possible. 2. For each of the matrices on the previous page tell whether or not the transformation is onto all of Rn. • If the matrix is NOT onto show this by finding a vector in R" that is not in the range of the transformation. If the matrix IS onto explain why this is not possible. Repeat 1 and 2 for each of the transformations described below. As part of this process, construct the standard matrix of the transformation and state what the domain ( R") and the codomain (R") are. T(x₁, x2 ) = ( 3x² + x2, 5x₁ + 7x2, x₁ + 3x₂ S(x1, x2, x3 ) = (x − 5x2 + 4x3, x2 - 6x3) R(x₁, X2, X3, X₁) = (0, x₁ + X2, X2 + X3, X3 +X₁) Q(x₁, x₂) = (x² + 4, x² + 5) (Q is a bit of a trick question. Why?)