Linear Transformations

Introduction

• A linear transformation is a function such that:
  • $f(u+v)=f(u)+f(v)$
  • $\alpha f(u)=f(\alpha u)$
• Prove a function is a linear transformation:
  • Prove it is true for all arbitrary vectors
• Disprove a function is a linear transformation:
  • Provide counterexample
• The zero vector must always be mapped to 0 for a linear transformation

Kernel

• The kernel of $f$ is all the vectors that are mapped to 0
• A kernel of a linear transformation is always a subspace

• The range of a linear transformation is also a subspace
  • You will need to be able to find the basis of the range and kernel

Review: Matrix Inversion and Determinants

Matrix Multiplication

• If $f(v)=Av$ and $v$ is a $n\times 1$ column vector (one column) then $f$ is a linear transformation
  • $A$ is some arbitrary $m\times n$ vector
• Every linear transformation is actually just matrix multiplication

Matrix of Transformation

• TBA

Invertible Matrices

• An arbitrary function $f:A\to B$ is called invertible if there is a function $g:B\to A$ such that $f^{-1}(g(x))=x$

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