Linear Transformations

$T:V\to W$ $T$ is a linear transform from V (domain) to W (codomain)

Conditions

$\forall x,y\in V$ and $c\in F$

• $T(x+y)=T(x)+T(y)$
• $T(cx)=cT(x)$

Low Dim Visual Meaning

• The origin stays in place
• All lines (in all directions) remain lines (they don’t curve)
  • Keeps lines parallel and equally spaced

Additional Properties

• Linear $⟹T(0)=0$
• Linear $⟺T(cx+y)=cT(x)+T(y)$
  • Generalization: $T({\sum }_{i=1}^n{a}_i{x}_i)={\sum }_{i=1}^n{a}_{i}T(x_i)$

Examples

• Rotation for a vector $(a_1,a_2)$ by $\theta$
• Reflection across x-axis
• Projection on x-axis
• $T(A)=A^t$
• Derivative of 1D function
• Certain types of integrals

Special Transforms

• Identity Transform: $I:V\to V$ defined by $I(x)=x,\forall x\in V$
• Zero Transform: $T_0:V\to W$ defined by $T_0(x)=0,\forall x\in V$

Notation

• $(T+U)(x)=T(x)+U(x)$
• $(aT)(x)=aT(x)$
• $UT=U(T)$ (because composition is equivalent to matrix multiplication)
• $\mathcal{L}(V,W)$ is the vector space of all linear transformations $V\to W$
  • Can be shortened to $\mathcal{L}(V)$ if $V=W$

Extra

• $T,U$ linear $⟹(aT+U)$ is linear
• The collection of all linear transformations $V\to W$ is a vector space over $F$

Nullspace and Range

Relationship with Bases

• A given linear transform is completely determined by its action on a basis
  • Can be thought of as mapping coordinates between bases
  • $T(a\cdot e_1+b\cdot e_2+\dots )=a\cdot e_1^{\prime }+b\cdot e_2^{\prime }+\dots$
• Theorem 2.6
  • $V,W$ vector spaces over $F$. $\{v_1,\dots ,v_n$} a basis for $V$
  • For $w_1,\dots ,w_n\in W$ there exists exactly one linear transformation $T:V\to W$ such that $T(v_i)=w_i$ for $i=1,\dots ,n$
• Corollary
  • If $U,T:V\to W$ are linear and $U(v_i)=T(v_i)\forall i=1,\dots ,n$ then $U=T$

Matrix Representations

• All linear transformations have a unique matrix representation (Isomorphisms)
• Definition
  • $V,W$ finite dimensional vector spaces with ordered bases $\beta =\{v_1,\dots ,v_n\}$ and $\gamma =\{w_1,\dots ,w_m\}$ respectively
  • $T:V\to W$ linear
  • Then for each $1\le j\le n$, $\exists !a_{ij}\in F,1\le i\le m$ such that $T(v_j)={\sum }_{i=1}^m{a}_{ij}{w}_i,1\le j\le n$
    • Think about it like every basis vector of the domain gives a column
• Notated
  • $A=[T{]}_{\beta }^{\gamma }$ where $\beta$ is the domain basis and $\gamma$ is the codomain basis
  • If $V=W$ and $\beta =\gamma$ then notated $A=[T{]}_{\beta }$
• Allows for applying a transform to a vector by multiplying the matrix by the vector
• Properties
  • $[T+U{]}_{\beta }^{\gamma }=[T{]}_{\beta }^{\gamma }+[U{]}_{\beta }^{\gamma }$
  • $[aT{]}_{\beta }^{\gamma }=a[T{]}_{\beta }^{\gamma }$
  • $[T(u){]}_{\gamma }=[T{]}_{\beta }^{\gamma }[u{]}_{\beta }$

Composition

• Equivalent to matrix multiplication of the two matrix representation of the transforms
  • $V,W,Z$ have ordered bases $\alpha ,\beta ,\gamma$ respectively
  • Let $T:V\to W$ and $U:W\to Z$ be linear
  • Then $[UT{]}_{\alpha }^{\gamma }=[U{]}_{\beta }^{\gamma }[T{]}_{\alpha }^{\beta }$
• $T,U$ linear $⟹(UT)$ is linear
• Let $V$ vector space; $T,U_1,U_2\in \mathcal{L}(V)$
  • $T(U_1+U_2)=T{U}_1+T{U}_2$ and $(U_1+U_2)T=U_{1}T+U_{2}T$
  • $T(U_1{U}_2)=(T{U}_1)(U_2)$
  • $TI=IT=T$
  • $a(U_1{U}_2)=(a{U}_1)U_2=U_1(a{U}_2),\forall a\in F$