Linear Transformations

In this section:

• $V$ and $W$ are vector spaces over $\mathbb{F}$
• $\dim V=n$ and $\dim W=m$
• $\mathcal{B}=(v_1,\dots ,v_n)$ and $\mathcal{C}=(w_1,\dots ,w_m)$ are bases of $V$ and $W$. (resp.)

Linearity

• Given a function $T:V\to W$. The following statements are equivalent:

  • $T$ is a linear transformation
  • (9.1.1) $T$ is additive and homogeneous
    • Additivity: $T(v_1+v_2)=T(v_1)+T(v_2)$
    • Homogeneity: $\alpha T(v)=T(\alpha v)$
  • (9.1.3) $T({\lambda }_1{v}_1+{\lambda }_2{v}_2)={\lambda }_{1}T(v_1)+{\lambda }_{2}T(v_2)$
  • $\exists A\in M_{m\times n}(\mathbb{F}):\forall v\in V,T(v)=Av$

• Theorems

  • (9.1.2a) If $T(0_V)\ne 0_W$, then $T$ is not linear
  • (9.7.1) let $S$ and $T$ linear transformations, then $f(v)=S(v)+T(v)$ is also linear transformation
  • $T(v)=T(u)⟺T(v-u)=0$

Hom

• $\text{Hom}(V,W)$ is the set of all linear transformations from $V$ to $W$
• (9.7.6) $\text{Hom}(V,W)$ is a vector space
  • $\text{Hom}(V,W)$ and $M_{m\times n}$ are isomorphic vector spaces
  • $\dim \text{Hom}(V,W)=m\times n$
  • Vector space operations: (see q9.7.2-3)
    • Addition: $(T+S)(v)=T(v)+S(v)$
      • Commutativity: $T+S=S+T$
      • Associativity: $(T_1+T_2)+T_3=T_1+(T_2+T_3)$
      • Identity: $T+0=T$
      • Inverse: $T+(-T)=0$
    • Scalar multiplication:
      • Distributivity (vector (transformations) addition): $\alpha (T+S)=\alpha T+\alpha S$
      • Distributivity (field addition): $(\alpha +\beta )T=\alpha T+\beta T$
      • Compatibility: $\alpha (\beta T)=(\alpha \beta )T$
      • Identity: $1T=T$
  • Vector space properties of $\text{Hom}(V,W)$
    • $T-S=0⟹T=S$
    • $T+T=T⟹T=0$
    • $0\cdot T=0$
    • $\alpha T=0⟹\alpha =0$ or $T=0$

Composition

Equality

• (9.4.1) $T,S:V\to W$, and $B=\{v_1,\dots ,v_n\}$ spans $V$. then $T=S⟺\forall v\in B:T(v)=S(v)$

Transformation matrix

• (d10.1.1) The transformation matrix of the linear transformation $T:V\to W$, relative to the bases $\mathcal{B}$ and $\mathcal{C}$, is the $m\times n$ matrix (with entries in $\mathbb{F}$) defined by $A=[T{]}_C^B={\left[\begin{array}{ccc}|& & |\\ [T(v_1){]}_C& \cdots & [T(v_n){]}_C\\ |& & |\end{array}\right]}_{m\times n}$
• $A$ is the transformation matrix of $T$ by the bases $\mathcal{B}$ and $\mathcal{C}$ if and only if $\forall v\in V:T(v)=Av$
• $[T{]}_C^B[v{]}_B=[T(v){]}_C$

Linear Transformation

In this section:

• $T:V\to W$ is a linear transformation
• $A$ is the transformation matrix of $T$ by the bases $\mathcal{B}$ and $\mathcal{C}$

Fundamental Spaces

Image

• $\text{Im}(T)=\{T(v)\mid v\in V\}$ is column-space of $A$

• (9.3.6) let $T:V\to W$, and $\text{Sp}\{v_1,\dots ,v_n\}=V$, then, $\text{Im}T=\text{Sp}(\{T(v_1),\dots ,T(v_n)\})$

• (9.5.6) if $T$ is injective, and $K\subseteq V$ linearly indep., then $T(K)\subseteq W$ is also linearly indep.

Kernel

• $\ker (T)=\{v\in V:T(v)=0\}$ is equal to the null space of $A$

Rank

Surjective (Onto)

The following statements are equivalent:

• $T:V\to W$ is surjective (epimorphism)
• $A$ columns spans $W$
• (T is right-cancellable) $RT=ST⟹R=S$
• (T is right-invertible) There exists $S:W\to V$ such that $TS=I$
• $A$ has full row rank

Theorems:

• if $\dim V<\dim W$ then $T$ cannot be onto

Injective (One-to-One)

The following statements are equivalent:

• $T:V\to W$ is injective (monomorphism)
• (9.5.2) $\text{ker}(T)=\{0\}$
• $\dim (\ker T)=0$
• $T(u)=T(v)⟹u=v$
• The colmuns of $A$ are linearly independent
• (T is left-cancellable) $TR=TS⟹R=S$
• (T is left-invertible) There exists $S:W\to V$ such that $ST=I$
• $A$ has full column rank

Theorems: ($T:V\to W$ is injective)

• if $\dim W<\dim V$ then $T$ cannot be one-to-one
• (q9.6.3c) $v_1,\dots ,v_n\in V$ are linearly independent, if and only if, $T(v_1),\dots ,T(v_n)\in V$

Isomorphism

• (9.6.2) The following statements are equivalent
  • $T$ is an isomorphism (invertible linear transformation) from $V$ on $W$
  • $T$ is both injective and surjective. (bijective)
  • $A$ has both a full row rank and a full column rank

dimV=dimW

In this section, $\dim V=\dim W$ ($m=n$), that is $A$ is a square matrix.

Theorems

• (9.4.2) Let $B=(v_1,\dots ,v_n)$ be a basis of $V$ and $(w_1,\dots ,w_n)$ an arbitrary list of vectors in $W$. Then there exists a unique linear map $T:V\to W$ such that $T(v_i)=w_i$

Isomorphism

• (9.6.2) The following statements are equivalent
  • $T$ is a linear isomorphism (bijective linear transformation) from $V$ on $W$
  • $T$ is injective
  • $T$ is surjective
  • $T$ is bijective
  • There is $T^{-1}:W\to V$ such that $T^{-1}T=I_V$
  • There is $T^{-1}:W\to V$ such that $T{T}^{-1}=I_W$
  • (10.5.1-2) $A=[T{]}_C^B$ is invertible, i.e. $A^{-1}=[T^{-1}{]}_B^C$ exists
  • (9.9.2) the inverse $T^{-1}:W\to V$ exists, such that $T^{-1}T=I_V$ and $T{T}^{-1}=I_W$

Linear Endomorphism $T:V\to V$

In the case where $V=W$, a linear map is called a linear endomorphism. Sometimes the term linear operator refers to this case, but the term linear operator can have different meanings for different conventions (wikipedia)

• $T^2=0⟹\mathrm{Im}T\subseteq \mathrm{Ker}T$

Eigenvalues

Equivalent definitions of eigenvalue for the linear transformation $T:V\to V$.

• (d11.2.1) $\lambda$ is an eigenvalue of $T$
• (d11.2.1) There exists a non-zero vector $\mathbf{v}$ in $V$ such that $T(\mathbf{v})=\lambda \mathbf{v}$.
• (In such a case, $\mathbf{v}$ is called an eigenvector of $T$ related to the eigenvalue $\lambda$)
• The operator $(T-\lambda I)$ is singular.
• $\text{rank}(T-\lambda I)<\dim (V)$
• $(\lambda I-T)(\mathbf{v})=0$ has nontrivial solutions, i.e., $\text{null}(\lambda I-T)\ne \{0\}$
• (11.4.1) The characteristic equation $\det (\lambda I-T)=0$
• $\lambda$ is a root of the characteristic equation $\det (xI-T)=0$

Theorems:

• (q11.2.4a) if $\lambda$ is an eigenvalue of $T$, then for each $\mu$, $\mu \lambda$ is an eigenvalue of $\mu T:V\to V$
• (q11.2.4b) if $\lambda$ is an eigenvalue of $T$, then , ${\lambda }^k$ is a eigenvalue of $T^k:V\to V$. (for each natural $k$ )
• (11.2.6) $T$ has at most $n$ distinct eigenvalues
• if $T^n=I$ for some natural $n$, then $T$ has at most the eigenvalues $1,-1$ (todo by q11.2.4)

Eigenvectors

Definitions of eigenvector. The following statements are equivalent:

• (d11.2.1) $\mathbf{v}$ is an eigenvector of $T$ related to $\lambda$.
• (d11.2.1) $\mathbf{v}$ is a non-zero vector in $V$ such that $T(\mathbf{v})=\lambda \mathbf{v}$.

Eigenspace

Definitions of the eigenspace of $T$ associated with its eigenvalue $\lambda$.

• (d11.2.2) $V_{\lambda }=\{\mathbf{v}\in V\mid T(\mathbf{v})=\lambda \mathbf{v}\}$
• $\{\text{eigenvectors corresponding to }\lambda \}\cup \{0\}$

Diagonalizability

• (d11.1.1) $T:V\to V$ is diagonalizable

• $V$ has a basis such that the matrix of $T$ (by that basis), is diagonal

• (11.2.3) $V$ has a basis in which all its vectors are eigenvectors of $T$

• (11.2.5) if $T$ has $n$ distinct eigenvalues, then $T$ is diagonalizable #todo https://textbooks.math.gatech.edu/ila/diagonalization.html

characteristic polynomial

• (d11.4.4) the characteristic polynomial of $T:V\to V$ is the characteristic polynomial of the transformation matrix by some basis
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