Solve problems using Frameworks

Understanding Frameworks in Applied Mathematics

You should understand and apply the principles and methods that underpin frameworks in applied mathematics.
You’ll need to be familiar with the basics of vectors, matrices and linear transformations.
Geometric interpretation of matrices as transformations in 2-D or 3-D space is very important.
Understanding how different transformations (e.g., translation, rotation, reflection, dilation) can be represented by matrices is crucial.

Solving Problems using Matrices

Practise solving equations by applying row operations to reach row-reduced echelon form.
Understand the connection between the rank of a matrix and its solutions.
Get comfortable with finding the inverse and determinant of matrices and how to use these when solving systems of equations.
Understanding the properties of matrices (commutativity, associativity, distributivity, identity and inverse) will help in various operations with matrices.

Working with Linear Transformations

Know how matrices can represent linear transformations and why those transformations are valuable.
Master matrix multiplication, especially when one of the matrices represents a transformation.
Have a clear understanding of vectors in 2 or 3 dimensions so you can better understand the effects of matrix transformations on them.

Vector Spaces and Related Concepts

You’ll need to understand vector spaces, subspaces, and the basis and dimension of a vector space.
Learn about linear independence and dependence, and how to determine whether a set of vectors is linearly independent or dependent.
Understand how to find a basis for a vector space, and know the consequences of a set of vectors spanning a space.

Eigenvalues and Eigenvectors

Understand what eigenvalues and eigenvectors are, and why they’re useful.
Get comfortable calculating eigenvalues and eigenvectors for a given square matrix.
Understand how to use eigenvalues and eigenvectors in a range of applications, including shifting basis and principal components analysis.

Remember, rigorous practice is the key to building your confidence with these concepts and operations. With this solid foundation, you will be better prepared to solve problems using frameworks in applied mathematics.