Linear Transformations

Linear-Transformations

1.1. Linear Transformation

1.1.1. Linear Transformation

  • Definition: A mapping L from a vector space V into a vector space W is said to be a linear transformation if \(L(\alpha v_1+ \beta v_2 )= \alpha L(v_1 )+ \beta L(v_2 )\) for all $v_1,v_2\in V$ and for all scalars 𝛼 and 𝛽.
  • Alternative definition: L is a linear transformation if and only if for all $v_1,v_2\in V$ and scalars 𝛼, \(L(v_1+v_2) = L(v_1) + L(v_2)\) \(L(\alpha v) = \alpha L(v)\)

1.1.2. Mapping

  • Definition: A mapping L from a vector space V into a vector space W will be denoted \(L : V \rightarrow W\) When the arrow notation is used, it will be assumed that V and V represent vector spaces.

1.1.3 linear operator

  • Definition: When the vector spaces V and W are the same, we will refer to a linear transformation L : V \rightarrow V as a linear operator on V. Thus, a linear operator is a linear transformation that maps a vector space V into itself.

L𝑖n𝑒𝑎𝑟 𝑜𝑝𝑒𝑟𝑎𝑡𝑜𝑟 ⊂ 𝑙𝑖n𝑒𝑎𝑟 𝑡𝑟𝑎n𝑠𝑓𝑜𝑟m𝑎𝑡𝑖𝑜n \(L:V\rightarrow V \subset L:V\rightarrow W\)

1.1.4. Linear Transformations from $\mathbb{R}^n$ to $\mathbb{R}^m$

  • Definition: If A is any m×n matrix, and define \(L:\mathbb{R}^n\rightarrow \mathbb{R}^m\) via \(L(x)=Ax\) for all $x\in\mathbb{R}^n$. Then L is a linear transformation from $\mathbb{R}^n$ to $\mathbb{R}^m$.
  • Property: If L is a linear transformation mapping a vector space V into a vector space W, then
    • $L(O_V) =O_W$ (where $O_V$ and $O_W$ are the zero vectors in V and W, respectively).
    • If $v_1,…,v_n$ are elements of V and $\alpha_1,…,\alpha_n$ are scalars, then $L(\alpha_1 v_1+\alpha_2 v_2+…+\alpha_n v_n)$=$\alpha_1 L(v_1 )+\alpha_2 L(v_2 )…+\alpha_n L(v_n)$
    • $L(−v)= −L(v)\ \forall v\in V$.

1.1.5. Kernel

  • Definition: Let L : $V \rightarrow W$ be a linear transformation. The kernel of L, denoted ker⁡(L), is defined by
    • $ker⁡(L)$ = {$v\in V L(v) = O_W$}
  • The kernel of L is the set of all vectors from V that L transforms to the zero vector in W.
  • Note, if L is defined via matrix multiplication for some matrix A, then ker⁡(L)=𝑁(A), the null space of A.

1.1.6. Image

  • Definition: Let L : V \rightarrow W be a linear transformation and let S be a subspace of V. The image of S, denoted L(S), is defined by $L(S)$ = {$w\in W |w= L(v)$ for some $ v\in S$}

1.1.7. Range

  • Definition: The image of the entire vector space, L(V), is called the range of L.

In other word

  • If {$v_1, v_2,…,v_k$} is a spanning set of S, then L(S) is the span of $L(v_1), L(v_2),… L(v_k)$.
  • If L is defined via matrix multiplication for some matrix A, then L(V), the range of L, is the column space of A.

1.1.8. Theorem 2.1.1 (Kernel/Image are subspaces)

If L : V \rightarrow W is a linear transformation and S is a subspace of V. Then

  • $ker⁡(L)$={$v\in V:L(v)=O_W$} is a subspace of V.
  • $L(S)$={$w\in W:w=L(v)$ for some $v\in S$} is a subspace of W.

1.2. Matrix Representations of Linear Transformations

1.2.1. Theorem 2.2.1 (Standard matrix representation)

If L is a linear transformation mapping $\mathbb{R}^n$ into $\mathbb{R}^m$, then there is an m×n matrix A, such that for each $x\in\mathbb{R}^n$, L(x)=Ax. In fact, the $j^{th}$ column vector of A is given by \(a_j=L(e_j)\)(j=1,2,…,n)

  • How to write the m×n matrix A:
    • See what L does to the basis vectors $v_1, v_2,…,v_n$ in V. Because $L(v_1)$ is in W, write it as linear combination of $w_1, w_2,…,w_m$;
    • write the coordinate vector $[L(v_1)]_𝐹$ as the first column of matrix A.
    • Repeat for $L(v_2),L(v_3)…$, to find the $2{nd}, 3{rd}$,… column of A.

1.2.2. Theorem 2.1.2

If 𝐸={$v_1,v_2,…,v_n$} and 𝐹={$w_1,w_2,…,w_m$} be ordered bases for $\mathbb{R}^n$ and $\mathbb{R}^m$, respectively. If L: $\mathbb{R}^n\rightarrow \mathbb{R}^m$ is a linear transformation and A is the matrix representing L w.r.t. 𝐸 and 𝐹, then \(a_j=W^{−1} L(𝒗_j )\) (j=1,2,…,n) Where W=$[w_1,w_2,…,w_m]$.

1.2.3. Corollary

If A is the matrix representing the linear transformation L: $\mathbb{R}^n\rightarrow \mathbb{R}^m$ with respect to the bases 𝐸={$u_1,u_2,…,u_n$} and 𝐹={$b_1,b_2,…,b_m$} then the reduced row echelon form of [$b_1,b_2,…,b_m$|$L(u_1),…,L(u_n)$] is $[𝐼|A]$. ___

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