Linear-1 9

Summary of previous topics

\begin{matrix} Let A \in F^{n \times n} \\ \exists A^{- 1} \\ ⟺ C F (A) = I \\ ⟺ r a n k (A) = n \\ ⟺ Columns of A are a linear independence \\ ⟺ Rows of A are a linear independence \\ ⟺ N (A) = {0} \\ ⟺ \exists! x : A x = 0 \\ ⟺ \exists b : \exists! x : A x = b \\ ⟺ \forall b : \exists! x : A x = b \end{matrix}

\begin{matrix} V, U - vector spaces over F \\ Function T : V \to U is called a linear transformation \\ If \forall v_{1}, v_{2} \in V, α \in F : T (v_{1} + α v_{2}) = T (v_{1}) + α T (v_{2}) \end{matrix}

\begin{matrix} T : R \to R \\ T (x) = x^{2} \\ T (2 + 2) = T (4) = 16 \\ T (2) + T (2) = 4 + 4 = 8 \\ ⟹ T is not a linear transformation \end{matrix}

\begin{matrix} Let T : V \to U be a linear transformation \\ ⟹ T (0_{V}) = 0_{U} \\ Proof: \\ T (0_{V}) = T (0_{F} \cdot 0_{V}) = 0_{F} \cdot \underset{\in U}{\underset{⏟}{T (0_{V})}} = 0_{U} \end{matrix}

\begin{matrix} V, U - vector spaces over F \\ B = {v_{1}, \dots, v_{n}} - basis of V \\ Let u_{1}, \dots, u_{n} \in U \\ ⟹ Exists a unique linear transformation T : V \to U \\ such that: \forall i \in [1, n] : T (v_{i}) = u_{i} \\ Proof: \\ B is a basis of V ⟹ \forall v \in V : v = \sum_{i = 1}^{n} α_{i} v_{i} \\ Let T (v) = \sum_{i = 1}^{n} \underset{Coefficients from the linear combination of v}{\underset{⏟}{α_{i}}} u_{i} \\ Here should be some long proof that T is a linear transformation... \\ ⟹ T (\sum_{i = 1}^{n} α_{i} v_{i}) = \sum_{i = 1}^{n} α_{i} u_{i} \\ ⟹ T (v_{i}) = T (1 v_{i}) = 1 u_{i} = u_{i} \end{matrix}

\begin{matrix} T : R^{2} \to R^{3} \\ v_{1} = (\begin{matrix} 2 \\ 3 \end{matrix}), v_{2} = (\begin{matrix} 1 \\ 2 \end{matrix}) \\ T (v_{1}) = (\begin{matrix} 1 \\ 0 \\ 1 \end{matrix}), T (v_{2}) = (\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}) \\ {v_{1}, v_{2}} is a basis of R^{2} \\ (\begin{matrix} x \\ y \end{matrix}) = α v_{1} + β v_{2} \\ (\begin{array}{ccc} 2 & 1 & x \\ 3 & 2 & y \end{array}) \to {\begin{matrix} α = 2 x - y \\ β = 2 y - 3 x \end{matrix} \\ T (v) = T (\begin{matrix} x \\ y \end{matrix}) = (2 x - y) T (v_{1}) + (2 y - 3 x) T (v_{2}) = \\ = (2 x - y) (\begin{matrix} 1 \\ 0 \\ 1 \end{matrix}) + (2 y - 3 x) (\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}) = (\begin{matrix} y - x \\ 2 y - 3 x \\ y - x \end{matrix}) \end{matrix}

\begin{matrix} T : V \to U \\ S : U \to W \\ T, S are linear transformations \\ ⟹ (S \circ T) : V \to W is a linear transformation \\ Proof: \\ TODO by yourself \end{matrix}

\begin{matrix} I d : V \to V \\ \forall v \in V : I d (v) = v \end{matrix}

\begin{matrix} T : V \to U is a linear transformation \\ T is called invertible if exists linear transformation S : U \to V \\ such that \\ (S \circ T) = I d_{V} \\ (T \circ S) = I d_{U} \\ S is then called an inverse of T \end{matrix}

\begin{matrix} T is a linear transformation \\ T is invertible ⟺ T is bijective \end{matrix}

\begin{matrix} Linear transformation is also called homomorphism \end{matrix}

\begin{matrix} Invertible linear transformation \end{matrix}

\begin{matrix} Linear transformation from V to V \end{matrix}

\begin{matrix} Invertible endomorphism \end{matrix}

\begin{matrix} Injective linear transformation \end{matrix}

Surjective linear transformation