# General linear equations

**Linear mapping = linear transformation = linear function
**

Definition. Given vector spaces V

_{1}and V

_{2}, a

mapping L : V

_{1}→ V

_{2}is linear if

L(x + y) = L(x) + L(y),

L(rx) = rL(x)

for any x, y ∈ V

_{1}and r ∈ R.

**Basic properties of linear mappings:**

• L(r

_{1}v

_{1}+ · · · + r

_{k}v

_{k}) = r

_{1}L(v

_{1}) + · · · + r

_{k}L(v

_{k})

for all k ≥ 1, v

_{1}, . . . , v

_{k}∈ V

_{1}, and r

_{1}, . . . , r

_{k}∈ R.

• L(0

_{1}) = 0

_{2}, where 0

_{1}and 0

_{2}are zero vectors in

V

_{1}and V

_{2}, respectively.

• L(−v) = −L(v) for any v ∈ V

_{1}.

**Range and kernel**

Let V,W be vector spaces and L : V → W be a

linear mapping.

Definition. The

**range**(or

**image**) of L is the set

of all vectors w ∈ W such that w = L(v) for some

v ∈ V. The range of L is denoted L(V).

The kernel of L, denoted ker L, is the set of all

vectors v ∈ V such that L(v) = 0.

**Theorem (i)**The range of L is a subspace of W.

**(ii)**The kernel of L is a subspace of V.

**General linear equations**

Definition. A linear equation is an equation of the form

L(x) = b,

where L : V → W is a linear mapping, b is a given vector

from W, and x is an unknown vector from V.

The range of L is the set of all vectors b ∈ W such that the

equation L(x) = b has a solution.

The kernel of L is the solution set of the homogeneous linear

equation L(x) = 0.

Theorem If the linear equation L(x) = b is solvable then the

general solution is

x

_{0}+ t

_{1}v

_{1}+ · · · + t

_{k}v

_{k},

where x

_{0}is a particular solution, v

_{1}, . . . , v

_{k}is a basis for the

kernel of L, and t

_{1}, . . . , t

_{k}are arbitrary scalars.

**Example.**

Linear equation: L(x) = b, where

**Example.**

Linear operator L :

Linear equation: Lu = b, where b(x) = e^{2x} .

It can be shown that the range of L is the entire

space C(R) while the kernel of L is spanned by the

functions sin x and cos x.

Observe that

By linearity,
is a particular solution.

Thus the general solution is

**Matrix transformations**

Any m×n matrix A gives rise to a transformation

L : R^{n} → R^{m} given by L(x) = Ax, where x ∈ R^{n}

and L(x) ∈ R^{m} are regarded as column vectors.

This transformation is linear.

Indeed, L(x + y) = A(x + y) = Ax + Ay = L(x) + L(y),

L(rx) = A(rx) = r (Ax) = rL(x).

**Example.
**

Let e_{1} = (1, 0, 0), e_{2} = (0, 1, 0), e_{3} = (0, 0, 1) be the

standard basis for R^{3}. We have that L(e1) = (1, 3, 0),

L(e_{2}) = (0, 4, 5), L(e_{3}) = (2, 7, 8). Thus L(e_{1}), L(e_{2}), L(e_{3})

are columns of the matrix.

Problem. Find a linear mapping L : R^{3} → R^{2}

such that L(e_{1}) = (1, 1), L(e_{2}) = (0,−2),

L(e_{2}) = (3, 0), where e_{1}, e_{2}, e_{3} is the
standard

basis for R^{3}.

Columns of the matrix are vectors L(e_{1}), L(e_{2}), L(e_{3}).

**Theorem Suppose** L : R^{n} → R^{m} is a linear map. Then

there exists an m×n matrix A such that L(x) = Ax for all

x ∈ R^{n}. Columns of A are vectors L(e_{1}), L(e_{2}),
. . . , L(e_{n}),

where e_{1}, e_{2}, . . . , e_{n} is the standard basis
for R^{n}.

**Linear transformations of R ^{2}**

Any linear mapping f : R

^{2}→ R

^{2}is represented as

multiplication of a 2-dimensional column vector by a

2×2 matrix: f (x) = Ax or

Linear transformations corresponding to particular

matrices can have various geometric properties.

Rotation by 90^{o} |

Rotation by 45^{o} |

Reflection in the vertical axis |

Reflection in the line x − y = 0 |

Horizontal shear |

Scaling |

Squeeze |

Vertical projection on the horizontal axis |

Horizontal projection on the line x + y = 0 |

Identity |