Preimage and kernel example | Matrix transformations | Linear Algebra | Khan Academy

TL;DR

This video explains how to find the preimage of a subset in the codomain under a linear transformation in R2 using matrix multiplication.

Transcript

Let's say I have some transformation from R2 to R2. And it's essentially just a multiplication times a matrix. And we know that all linear transformations can be expressed as a multiplication of a matrix, but this one is equal to the matrix 1, 3, 2, 6 times whatever vector you give me in my domain. Times x1, x2. Now let's say I have some subset in ... Read More

Key Insights

• ✖️ Linear transformations in R2 can be represented as matrix multiplication.
• ☺️ Preimages of subsets in the codomain can be found by solving A * x = b.
• 👾 The preimage is equivalent to the null space of the matrix A representing the transformation.
• 😫 The kernel of a transformation is the set of vectors that map to the zero vector.
• ☺️ The preimage of a subset can be represented as the solution set of the system of equations formed by A * x = b.
• 🫥 The preimage can be visualized as lines or sets of vectors in R2 that map to specific points in the codomain.

Questions & Answers

Q: How can linear transformations in R2 be represented?

Linear transformations in R2 can be represented as a multiplication of a matrix with a vector, i.e., A * x, where A is the transformation matrix and x is the vector in the domain.

Q: What is the preimage of a subset in the codomain?

The preimage of a subset in the codomain refers to all the vectors in the domain that, when transformed, map to the subset. It can be found by solving the equation A * x = b, where A is the transformation matrix and b is the subset vector.

Q: What is the relationship between the kernel of a transformation and the null space of a matrix?

The kernel of a transformation is the set of vectors in the domain that map to the zero vector in the codomain. This is equivalent to the null space of the corresponding matrix A representing the transformation.

Q: How are preimages of subsets in R2 determined?

Preimages of subsets in R2 are determined by finding the solutions to the equation A * x = b, where A is the transformation matrix, x is the vector in the domain, and b is the subset vector. This involves solving a system of linear equations using the augmented matrix [A | b].

Summary & Key Takeaways

• Linear transformations in R2 can be represented as a multiplication of a matrix with a vector.

• To find the preimage of a subset in the codomain, we need to solve the equation A * x = b, where A is the transformation matrix, x is the vector in the domain, and b is the subset vector.

• The preimage can be found by determining the null space of the matrix A, which represents all the vectors that when transformed, result in the desired subset.