04 Linear Transformations
Linear Transformations¶
Consider a linear transformation
\[ L: \underbrace{U}_{\text{Domain}} \to \underbrace{W}_{\text{Codomain}} \]
Properies¶
- \(L(O_u) = O_w\)
- \(L(\vec u \oplus \vec v) = L(\vec u) + L(\vec v)\)
- \(L(\alpha \odot u) = \alpha \cdot L(\vec u)\)
Tricks¶
A transformation is not Linear Transformation if
- Power \(\ne\) 1 or 0
- there is modulus(absolute value)
- determinant
Kernel¶
\[ S = \set{ \vec u: L(\vec u) = O_w } \]
Set of all input values
Range¶
\[ S = \set{ L(\vec u) } \]
Set of all output values
Properties¶
| Property | Condition |
|---|---|
| One-one | Kernel = \(\set{O_u}\) |
| Onto | dim(range) = dim(codomain) |
Dimension Theorem¶
\[ \text{ dim(range) + dim(kernel) = dim(U) } \]