📚 Topic Summary
When we have a linear transformation, we can represent it as a matrix. However, this matrix representation depends on the choice of bases for the vector spaces involved. Changing the bases changes the matrix. So, finding the matrix representation involves expressing the transformation's effect on the basis vectors of the domain in terms of the basis vectors of the codomain. This exercise focuses on precisely that: given different bases, how do we find the corresponding matrix representation of a linear transformation? It reinforces the understanding of how the same linear transformation can have different matrix forms depending on the perspective (i.e., the chosen bases).
Deriving matrix representations with arbitrary bases allows us to analyze linear transformations in coordinate systems that are best suited to the problem at hand. This is particularly useful in fields like computer graphics, where transformations like rotations and scaling are frequently expressed using different coordinate systems.
🧠 Part A: Vocabulary
Match the term to its definition:
| Term |
Definition |
| 1. Linear Transformation |
A. A set of linearly independent vectors that span the entire vector space. |
| 2. Basis |
B. A mapping between two vector spaces that preserves vector addition and scalar multiplication. |
| 3. Matrix Representation |
C. The vector space that contains all possible outputs of a linear transformation. |
| 4. Domain |
D. A rectangular array of numbers arranged in rows and columns used to represent a linear transformation with respect to specific bases. |
| 5. Range |
E. The vector space on which a linear transformation is defined. |
Match the correct term to its definition.
✍️ Part B: Fill in the Blanks
A ______ transformation is a function between two vector spaces that preserves vector addition and scalar multiplication. The choice of ______ affects the _______ representation of a linear transformation. The set of all possible output vectors is known as the ______. To find the matrix, express how the linear transformation affects each ______ vector in the domain.
🤔 Part C: Critical Thinking
Explain why the matrix representation of a linear transformation changes when different bases are used. Provide an example to illustrate your explanation.
