📖 Theorem 10.
Let \(T : \mathbb{R^n} \to \mathbb{R^m}\) be a linear transformation. Then there exists a unique matrix A such that
\(T(\mathbf{x}) = A\mathbf{x} \mbox{ for all }\mathbf{x} \mbox{ in }\mathbb{R^n}\)
In fact, \(A\) is the \(m \times n\) matrix whose j-th column is the vector \(T(e_j)\), where \(e_j\) is the j-th column of the identity matrix in \(\mathbb{R^n}\):
\(A = \left[ T(e_1) \cdots T(e_n) \right]\)
Standard matrix for the linear transformation T
Onto
A mapping \(T: \mathbb{R^n} \to \mathbb{R^m}\) is said to be onto \(\mathbb{R^m}\) if each \(\mathbf{b}\) in \(\mathbb{R^m}\) is the image of at least one \(\mathbf{x}\) in \(\mathbb{R^n}\)
One-to-one
A mapping \(T: \mathbb{R^n} \to \mathbb{R^m}\) is said to be one-to-one if each \(\mathbf{b}\) in \(\mathbb{R^m}\) is the image of at most one \(\mathbf{x}\) in \(\mathbb{R^n}\)
- 여러 개의 벡터에서 같은 이미지로 도달할 때 : one-to-one이 아님
- 1대1 대응 : one-to-one
📖 Theorem 11.
Let \(T: \mathbb{R^n} \to \mathbb{R^m}\) be a linear transformation.
Then \(T\) is one-to-one if and only if the equation \(T(\mathbf{x}) = 0\) has only trivial solution.
\(T(0) = T(0\mathbf{0}) = 0T(\mathbf{0})\\
T(\mathbf{u} = \mathbf{b} T(\mathbf{v} = b)\\
T(\mathbf{u} - \mathbf{v}) = T(\mathbf{u}) - T(\mathbf{v}) = 0\\
\mathbf{u} - \mathbf{v} \ne 0\)
\(T(\mathbf{x}) = 0\) has more than one solution
📖 Theorem 12.
Let \(T: \mathbb{R^n} \to \mathbb{R^m}\) be a linear transformation and let \(A\) be the standard matrix for \(T\).
Then \(T\) maps \(\mathbb{R^n} \mbox{onto} \mathbb{R^m}\) if and only if the columns of \(A\) span \(\mathbb{R^m}\)
\(T\) is one-to-one if and only if the columns of \(A\) are linearly independent.
one-to-one → only trivial → independent