Linear Transformations
Part 2: Injectivity, Surjectivity and Isomorphisms
We begin with two definitions.
A transformation T from a vector space V to a vector space W is called injective (or onetoone) if T(u) = T(v) implies u = v. In other words, T is injective if every vector in the target space is "hit" by at most one vector from the domain space.
A transformation T mapping V to W is called surjective (or onto) if every vector w in W is the image of some vector v in V. [Recall that w is the image of v if w = T(v).] Alternatively, T is onto if every vector in the target space is hit by at least one vector from the domain space.
 Let T_{A} be the linear transformation defined by multiplication by the matrix A from Part 1. Enter the matrix A and the vectors u_{1} and u_{2}. Find the images of u_{1} and u_{2} under T_{A}. What can you say about T_{A}?
 Describe the set of all vectors x such that T_{A}(x) = (2, 1, 3). [Hint: You must solve a matrix equation of the form Ax = b.]
 Is the vector w = (1, 1, 2) in the image of T_{A}? Explain how you know. Is T_{A} surjective?
 Let T_{C} be the linear transformation defined by multiplication by the matrix C defined in your worksheet. Enter C. Find the vector u such that T_{C}(u) = (5, 4, 1). How do you know there is only one possibility for u?
 Explain why T_{C} must be an injective map.
 Complete the following statement:
A linear transformation T, defined by x > Ax, is injective if and only if the matrix equation Ax = b has ...
(describe the number of solutions for each possible b).
 Complete the following statement:
A linear transformation T, defined by x > Ax, is surjective if and only if the matrix equation Ax = b has ...
(describe the number of solutions for every possible b).
A linear transformation T from a vector space V to a vector space W is called an isomorphism of vector spaces if T is both injective and surjective.
 If a linear transformation is an isomorphism and is defined by multiplication by a matrix, explain why the matrix must be square. What else is special about the matrix?
 Create a 3 x 3 matrix M which defines an isomporphism from R^{3} to R^{3}. Find a matrix which defines the inverse of the original linear transformation.
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