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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 one-to-one) 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.

  1. Let TA be the linear transformation defined by multiplication by the matrix A from Part 1. Enter the matrix A and the vectors u1 and u2. Find the images of u1 and u2 under TA. What can you say about TA?

  2. Describe the set of all vectors x such that TA(x) = (2, 1, 3). [Hint: You must solve a matrix equation of the form Ax = b.]

  3. Is the vector w = (1, 1, 2) in the image of TA? Explain how you know. Is TA surjective?

  4. Let TC be the linear transformation defined by multiplication by the matrix C defined in your worksheet. Enter C. Find the vector u such that TC(u) = (5, 4, -1). How do you know there is only one possibility for u?

  5. Explain why TC must be an injective map.

  6. 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).

  7. 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.

  1. 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?

  2. Create a 3 x 3 matrix M which defines an isomporphism from R3 to R3. Find a matrix which defines the inverse of the original linear transformation.

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