Sub-Spaces

SUB-SPACES

Definition: Sub-spaces

A subset W of a vector space V over a field F is called a sub-space of V if W is a vector space over F with the operations of addition and scalar multiplication defined on V.

Note

1. In any vector space V, V and {0} are subspaces

2. {0} is called the zero subspace of V.

3. A subset W of a vector space V is a sub-space of V if and only if the following four properties holds

(a) x + y ε W whenever x ε W and y ε W

(b) c x ε W whenever c ε F and x ε W

(c) W has a zero vector.

(d) Each vector in W has an additive inverse in W.

Example :

Consider 

It is a vector space for addition and scalar multiplication of vectors.

Consider 

We can easily see that it is a vector space for addition and scalar multiplication of vectors.

Further W C V

.'. W is a vector sub-space of V.

(a) Function subspace Fⁿ

Theorem:

Let V be a vector space and W a subset of V. Then W is a subspace of V if and only if the following three conditions hold for the operations defined in V.

Proof :

Given: V is a vector space and W is a subset of V.

If part: W is a subspace of V then prove that (a), (b) & (c) conditions.

Proof : By definition of W is a subspace of V, (b) and (c) holds

So condition (a) holds

Converse part :

To prove W is a sub-space of V

If the conditions (a), (b) & (c) hold only, the remainder is each vector in W has an additive inverse in W.

Thus the additive inverse of each element of W is also in W.

Hence, W is a sub-space of V.

Problem 1.

Prove that a subset W of a vector space V is a sub-space of V if and only if 0 ε W and ax + y ε W whenever a ε F and x, y ε W

Solution :

Given: V is a vector space and W is a sub-set of V.

If part: If W is a sub-space of V then

(i) 0 ε W

(ii) ax + y ε W whenever a ε F and x, y ε W

Proof : W is a sub-space of V

So conditions (i) & (ii) hold

Converse part :

Given:

i.e., To Prove :

Thus the additive inverse of each element of W is also in W.

Hence, W is a sub-space of V.

Problem 2.

Solution :

Problem 3

Solution :

(b) Matrix sub-space

Example 1 :

An n x n matrix, M is called a diagonal matrix if M_ij = 0 whenever i ≠ j, that is, if all its non-diagonal entries are zero.

Clearly the zero matrix is a diagonal matrix because all of its :noitulo2 entries are 0.

Moreover, if A and B are diagonal n x n matrices, then whenever i ≠ j

 for any scalar c.

Hence, A + B and c A are diagonal matrices for any scalar c. Therefore the set of diagonal matrices is a subspace of M_{n x n}(F)

Example 2.

The trace of an n x n matrix M, denoted tr(M), is the sum of the diagonal entries of M; that is,

It follows that the set of n x n matrices having trace equal to zero is a sub-space of M_{n x n}

Example 3.

The set of matrices in M_{m x n}(R) having non-negative entries is not a sub space of M_{m x n}(R) because it is not closed under scalar multiplication (by negative scalars).

Problem 1.

Determine the transpose of the matrices. In addition, if the matrix is square, compute its trace.

Solution :

Problem 2

Solution :

Problem 3

Prove that diagonal matrices are symmetric matrices.

Solution:

If A is a diagonal matrix,

we have A_jj = 0 = A_ji when i ≠ j

⇒ A is symmetric.

(c) Intersection of sub-spaces

Theorem :

Any intersection of sub-spaces of a vector space V is a sub-space of V.

Proof :

Let C be a a collection of sub-spaces of V.

Let W = ∩ C

Clearly W is not empty as 0 ε ∩ C

Problem 1.

The intersection of the sub-spaces W₁ and W₂ of a vector space V is also a sub-space.

Solution :

Problem 2.

Prove that the intersection of any two sub-sets of V is not a sub-space of V.

Solution :

Problem 3.

Prove that the union of two sub-spaces of a vector space need not be a sub-space.

Solution :

Hence, W is a sub-space of R³

However A U B is not a sub-space of R³

Problem 4.

Let W₁ and W₂ be sub-spaces of a vector space V. Prove that 

(OR)

The union of two sub-spaces is a sub-space iff one is contained in the other. [A.U. N/D 2019, R-17]

Solution:

Let W₁ and W₂ be sub-spaces of a vector space V.

Ist Part :

Proof:

Converse Part :

To prove : 

Proof : Let us assume that

Which is a contradiction.

This gives a contradiction.

(d) Vector sub-space R³

Definition: A subset W of a vector space V is a subspace of V if it satisfies.

Problem 1.

Solution :

All the three conditions are satisfied.

Hence, W is a sub-space of V.

Problem 2.

Solution :

Problem 3.

Show that the set

Solution :

All the three conditions are satisfied.

.'. W is a subspace of V.

Problem 4.

Show that the set

Solution :

(e) Function sub-space 

Problem 1.

Let S be a non-empty set and F a field. Prove that for any  is a sub-space of

Solution :

And zero function is in the set.

.'. Given is a sub-space of 

Problem 2.

Let S be a non-empty set and F a field. Let C(S, F) denote the set of all functions f ε  such that f(s) = 0 for all but a finite number of elements of S. Prove that C (S, F) is a sub-space of .

Solution :

The number of non-zero points of f + g is less than the number of union of non-zero points of ƒ and g.

=> It is closed under addition.

The number of non-zero points of c f equals to the number of f.

=> It is closed under scalar multiplication.

And zero function is in the set.

.'. C(S, F) is a sub-space of 

 

Problem 3.

Is the set of all differentiable real-valued functions defined on R a sub-space of C(R)? Justify your answer.

Solution :

The sum of two differentiable functions and product of one scalar and one differentiable function are again differentiable.

The zero function is differentiable.

.'. Yes, the set of all differentiable real-valued functions defined on R a sub-space of C(R).

(f) Sequence and Polynomial sub-space F.

Problem 1.

Show that the set of convergent sequence {a_n} is a sub-space of the vector space

Solution :

We know that, {a_n + b_n} and {t a_n} will converge. And zero sequence, that is sequence with all entries zero, will be the zero vector.

Therefore the set of convergent sequence {a_n} is a sub-space of the vector space V.

Problem 2.

Is the set  a sub-space of P(F) if n ≥ 1 ? Justify your answer.

Solution :

No in general, but yes when n = 1,

Since, W is not closed under addition, when n =  is not in W.

Definition:

If S₁ and S₂ are non-empty sub-sets of a vector space V, then the sum of S₁ and S₂, denoted S₁ + S₂, is the set 

Definition:

A vector spave V is called the direct sum of W₁ and W₂ if W₁ and W₂ are sub-spaces of V such that W₁ ∩ W₂ = {0} and W₁ + W₂ = V. We denote that V is the direct sum of W₁ and W₂ by writing 

Note: The necessary and sufficient conditions for a vector space V to be the direct sum of two of its sub-spaces W₁ and W₂, are

Problem 1.

Let W₁ and W₂ be sub-spaces of a vector space V.

(a) Prove that W₁ + W₂ is a sub-space of V that contains both W₁ and W₂.

(b) Prove that any sub-space of V that contains both W₁ and W₂ must also contain W₁ + W₂.

Solution :

(a) By the definition of a subspace, if 0 is in W₁, 0 is in W₂

Thus, W₁ + W₂ satisfies the condition of a subspace.

Therefore, W₁ + W₂ is a subspace of V.

Problem 2.

Show that Fⁿ is the direct sum of the sub-spaces

Proof :

Given: W₁ is a sub-space of Fⁿ

Given: W₂ is also a sub-space of Fⁿ

=> Fⁿ is the direct sum of W₁ and W₂.

(h) Co-set of W containing V

Definition :

For any v Є V the set {v} + W = {v + w; w ε W} is called the co-set of W containing v.

Note: Denote this co-set v + W rather than {v} + W

Additional and scalar multiplication by scalars of F can be defined {v + W : v ε V} of all co-sets of W as follows :

Problem 1.

If W be a sub-space of a vector space V over a field F, prove that v + W is a sub-space of V if and only if v ε W.

Solution :

Part I. Given: v + W is a sub-space of V.

To prove v ε W

Proof: W is a sub-space, 0 ε W and so v + 0 = v ε v+W

=> v ε W

Part II. Given: v ε W

To prove: v + W is a sub-space of V.

Proof : Since W is a sub-space,

0 ε W

Hence, v + W is a sub-space.

Problem 2.

Let W be a sub-space of a vector space over a field F. Prove that 

Solution:

Part-I :

Proof: 

Part-II :

Proof : Consider the co-set v₂ + W

EXERCISE 4.2

1. Determine the transpose of each of the matrices given in addition if the matrix is square find its trace.

3. Let V denote the vector space consisting of all upper triangular n x n matrices and let W₁ denote the sub-space of V consisting of all diagonal matrices. Show that  where 

4. If W_i, 1 ≤ i ≤ 2 are vector sub-space of a vector space V, then W₁ ∩ W₂ is a vector sub-space.

5. If W_i, 1 ≤ i ≤ 2 be vector sub-spaces of a vector space V, when is W₁ U W₂ a sub-space of V?

6. Let W be a vector sub-space of V. What is w + W if w ε W? What is W + W? Is it true that w + W = W if and only if w ε W?

7. A sub-set W of a vector space V(F) is sub-space of V if and only if 

8. The union of two sub-spaces of a vector space V₃(R) may not be a sub-space of V₃(R)

9. Let V be the vector space of all square matrices over R. Determine which of the following are sub-spaces of V.

10. Let V be the space of all functions from R to R, and W₁ and W₂ be the sub-spaces of V defined by

11. Let W₁, W₂ be the sub-spaces of R³ given by