Invariant

(e) Invariant

Definitions :

Let V be a vector space, and let T : V → V be linear. A subspace W of V is said to be T-invariant if T(x) Є W for every x Є W, that is T(W) W. If W is T-invariant, we define the restriction of T on W to be the function T_W : W → W defined by T_W (x) = T(x) for all x Є W.

Problem 1.

If W is a subspace of a vector space V and that T : V → V is linear.

Prove that subspace {0}, V, R(T), and N(T) are all T-invariant.

Solution :

A subspace W of the vector space V is said to be T-invariant if T(x) Є W for every x Є W, that is T(W)  W.

If W is T-invariant, define the restriction of T on W be the function T_W : W → W defined by T_w(x) = T(x) for all x Є W, where T: V → V is linear.

Let the subspace W = {0}

Since T is linear, based on the property of linearity.

=> T(0) = 0, that is T(x) Є W

=> W = {0} is T-invariant by definition of T-invariant.

Let the subspace W = V.

Since T is defined as T : V → V, therefore T(V) V.

=> W = V is T-invariant by the definition of T-invariant.

Let the subspace W = R(T).

Let x Є R(T), that is some y Є V exists such that T(y) = x

Now, take the function T on both sides of T(y) = x, to get the following:

T(T(y)) = T(x)

Since T(y) Є V, T(x) Є R(T) and so, T(R(T)) C R(T)

W = R(T) is T-invariant by the definition of T-invariant.

Let the subspace W = N(T).

Let x Є N(T), that is T(x) = 0

W = N(T) is T-invariant by the definition of T-invariant.

Problem 2.

If W is T-invariant, prove that T_w is linear.

Solution :

To show that T_w is a linear if W is T-invariant.

A subspace W of the vector space V is said to be T-invariant if T(x) Є W for every x Є W, that is T(W)  W.

If W is T-invariant, define the restriction of T on W to be the function T_W  : W → W defined by T_W(x) = T(x) for all x Є W, where T : V → V is linear.

Let x, y Є W

Since W  V, so x,y Є V and for all x,y Є W

As T is linear on V,

. '.T(x + y) = T(x) + T(y)

T(α x) = α T(x), for α Є F

Thus, T is linear on W and hence T_W is linear.

Problem 3.

Suppose that T is the projection on W along some subspace W'. Prove that W is T-invariant and that T_W = I_W.

Solution :

To show that W is T-invariant and that T_W = I_W, if T is the projection on W along some subspace W'.

We know that, T : V → V is called the projection on W₁ along W₂ if, for x = x₁ + x₂ with x₁ Є W₁ and x₂ Є W₂.

We have T(x) = x₁, where W₁ and W₂ are the subspaces of the vector space V such that 

Since T is the projection on W along some subspace W', therefore using definition of projection

We have T(y) = y for all y Є W that is T(W)  W.

By the definition of T-invariant, => W is T-invariant.

Problem 4.

Suppose that V = R(T)  W and W is T-invariant.

Prove that W  N(T)

Solution :

Let T : V → V be a linear transformation.

Let W is a T-invariant subspace of V = R(T)  W.

Here, W is a T-invariant, then T(W)  W.

Here, W is a subspace of V => W C V

That is, for each w in W. T(W) = 0

Problem 5.

Suppose that V = R(T)  W and W is T-invariant.

(i) Show that if V is finite-dimensional, then W = N(T)

(ii) Show by example that the conclusion of (i) is not necessarily true if V is not finite-dimensional.

Solution :

(i) Let V is a finite dimensional vector space.

By Dimension Theorem,

(ii) Let V be a set of all polynomials over the field R is a vector space, then clearly V is not finite dimensional.

Define T(g(x)) = g'(x), for all g(x) Є V, then T is linear.

Now,

Therefore, the conclusion of part (i) is not necessarily true if V is not finite-dimensional.

Problem 6.

Suppose that W is T-invariant. Prove that N(T_W) = N(T) ∩ W and R(T_W) = T(W)

Solution :

Converse part: