10. The Matrix of a Linear Transformation
Dated: 18-03-2025
Theorem
Let \(T: \mathbb R^n \to \mathbb R^m\) to be a linear transformation1 then there exists a unique matrix2 \(A\) such that \(T(\vec x) = A \vec x \quad \forall \vec x \in \mathbb R^n\).
\(A\) is called the standard matrix for the linear transformation \(T\).
Existence and Uniqueness of the Solution of \(T(\vec x) = \vec b\)
Definition
Onto
A mapping1 \(T : \mathbb R^n \to \mathbb R^m\) is said to be onto \(\mathbb R^m\) if each \(\vec b\) in \(\mathbb R^m\) is the image of at least one \(\vec x\) in \(\mathbb R^n\).
One to One
A mapping1 \(T : \mathbb R^n \to \mathbb R^m\) is said to be onto \(\mathbb R^m\) if each \(\vec b\) in \(\mathbb R^m\) is the image of at most one \(\vec x\) in \(\mathbb R^n\).
Theorem
Let \(T : \mathbb R^n \to \mathbb R^m\) be a linear transformation.1 Then \(T\) is one to one if and only if the equation \(T(\vec x) = \vec 0\) has only the trivial solution.
Kernel of a Linear Transformation1
Let \(T : \mathbb V \to \mathbb W\) be a linear transformation1 then kernel of \(T\) (Kert \(T\)) is the set3 of those elements in \(\mathbb V\) which maps onto \(\vec 0\) in \(\mathbb W\).
The shaded area represents \(KerT\).
- \(KerT\) is subspace of \(\mathbb V\).
- \(T\) is one to one iff \(KerT = \vec 0\) in \(\mathbb V\).
Theorem
Let \(T : \mathbb R^n \to \mathbb R^m\) be a linear transformation1 and let \(A\) be the standard matrix for \(T\) then
- \(T\) maps \(\mathbb R^n\) onto \(\mathbb R^m\) if and only if the columns of \(A\) spans \(\mathbb R^m\).
- \(T\) is one to one if and only if the columns of \(A\) are linearly independent.
References
Read more about notations and symbols.