Linear Transformations

What is a linear transformation?

Let V and W be vector spaces over the same field (always R unless otherwise specified). A function T : V → W is called a linear transformation iff:

• T(u + v) = T(u) + T(v) for all vectors u, v in V

• T(λv) = λT(v) for all vectors v in V and all scalars λ

Equivalently, T : V → W is called a linear transformation iff:

• T(u + λv) = T(u) + λT(v) for all vectors u, v in V and all scalars λ

V is allowed to equal W

What is the zero transformation?

O(v) = 0

What is the identity transformation?

I_v(v) = v

What are some properties of linear transformations T : V → W?

• T(0_v) = 0_w

• T(-v) = -T(v) for all v in V

• T(λ₁v₁ + … + λ_kv_k) = λ₁T(v₁) + … + λ_kT(v_k) for all vectors v₁, …, v_k in V and scalars λ₁, …, λ_k

If V is n-dimensional with a basis B = {b₁, …, b_n} and there are 2 linear transformations T₁ : V → W and T₂ : V → W such that T₁(b_i) = T₂(b_i) for i = 1, 2, …, n then,…

T₁ = T₂. Moreover, for any vectors w₁, …, w_n in W, there exists a unique linear transformation T : V → W such that T(b_i) = W_i for all i = 1, 2, …, n

What is a composition?

Let T : U → V and S : V → W be linear transformations. Then the composition S ∘ T : U → W is the function (S ∘ T)(u) = S(T(u)) for all u in U

Prove this theorem: ‘Let T : U → V and S : V → W be linear transformations. Then the composition S ∘ T : U → W is a linear transformation.’

Take u₁, u₂ in U and scalar λ

(S ∘ T)(λu) = S(T(λu))

= S(λT(u))

= λS(T(u)) because S is a linear transformation

= λ(S ∘ T)(u)

So both criteria are satisfied, therefore S ∘ T is a linear transformation

What is the image?

Let T : V → W be a linear transformation. Then the image of T is the set of all possible outputs in W:

im(T) = {T(v) ∈ W : v ∈ V}

What is the kernel?

The kernel of T is the set of everything in V that gets mapped to 0_w:

ker(T) = {v ∈ V : T(v) = 0_w}

Prove this theorem: ‘Let T : V → W be a linear transformation. Then im(T) \< W and ker(T) \< V’

Take w₁, w₂ ∈ im(T)

Take scalar λ

We want w₁ + λw₂ ∈ im(T)

We know there exists v₁, v₂ ∈ V such that T(v₁) = w₁ and T(v₂) = w₂

Consider T(v₁ + λv₂) = T(v₁) + λT(v₂) = w₁ + λw₂

Since v₁ + λv₂ ∈ V, we’ve found a preimage of w₁ + λw₂ under T

Therefore, w₁ + λw₂ ∈ im(T)

Take v₁, v₂ ∈ ker(T) and scalar λ

We have T(v₁ + λv₂) = T(v₁) + λT(v₂) = 0 + 0λ = 0

Therefore v₁ + λv₂ ∈ ker(T)

Im(T) and ker(T) are both non-empty since T(0_v) = 0_w so 0_w ∈ im(T), 0_v ∈ ker(T)

What is the rank-nullity theorem?

Let T : V → W be a linear transformation. Then, dim(im(T)) + dim(ker(T)) = dim(V) (dim(im(T)) = rank of T, dim(ker(T)) = nullity of T)