Let V be a vector space over a field F≡R or C, where R is the set of real numbers and C is the set of complex numbers.
1. Definition: A non-empty subset W of V is called a subspace if it is a vector space with respect to the addition and scalar multiplication defined on V.
2. Characterization: A set W is a subspace of V if and only if
i. W ⊆ V
ii. 0 ∈ W
iii. x + y ∈ W; where x ∈ W , y ∈ W and
iv. α . x ∈ W; where x ∈ W , α ∈ F
3. Examples - I:
i. Every vector space V is a subspace of itself.
ii. {0} is a subspace of V.
iii. {0} is the only subspace of the vector space {0} over F.
4. Examples -II:
W₁={(x, 0, 0, …, 0) ∈ Rⁿ : x ∈ R} is a subspace of the vector space Rⁿ over R.
Reason: Clearly W₁⊆Rⁿ and (0, 0, 0…, 0)∈ W₁. Further, (x, 0, 0, …, 0)+ (y, 0, 0, …, 0)=(x+y, 0, 0, …, 0)∈W₁; where (x, 0, 0, …, 0) ∈W₁, (y, 0, 0, …, 0)∈W₁ and α.(x, 0, 0, …, 0)=(α.x, 0, 0, …, 0)∈W₁; where (x, 0, 0, …, 0)∈W₁, α∈F. Using the characterization given in Section2, we have the required result.
W₂={(x, x, 0) ∈ R³ : x ∈ R} is a subspace of the vector space R³ over R.
Reason: It is the same as discussed in Example(i) above.
W₃={(x, x, x) ∈ R³ : x ∈ R} is a subspace of the vector space R³ over R.
Reason: It is the same as discussed in Example(i) above.
Let M_{2 x 2} be the collection of 2x2 matrices with real entries. Then M_{2 x 2} forms a vector space over R with respect to matrix addition and matrix scalar multiplication and W₄={ $\left[\begin{array}{ccc} a & b\ c & 0 \end{array}\right]$ ∈ M_{2 x 2}: a, b, c ∈ R} is a subspace of the vector space M_{2 x 2}.
Reason: It is the same as discussed in Example(i) above.
W₅={(x+2, x, x) ∈ R³ : x ∈ R} is not a subspace of the vector space R³ over R in view of Section 2. it does not contains (0, 0, 0) which is the zero of the vector space R³ over R .
Thus it may be noted that a subset of a vector space need not to be its subspace.
5. Definitions: The subspaces V and {0} of V are called improper subspaces of V and subspaces other than V and {0} are called proper.
6. Subspaces of R: Subspaces of the vector space R over R are {0} and R only. That is,
(i) W ≡ {0} is a subspace of the vector space R over R,
(ii) W ≡ R itself is a subspace of the vector space R over R and
(iii) If W is a subspace of the vector space R over R, then W = {0} or R.
Proof:
(i) Clearly W ⊆ R and 0 ∈ W. Let x, y ∈ W and α ∈ R. Then x = y = 0. Hence x + y = 0 ∈ W and α . x = 0 ∈ W. Hence W ≡ {0} is a subspace.
(ii) Clearly W ⊆ R and 0 ∈ W. Let x, y ∈ R and α ∈ R. Hence x + y ∈ R and α . x ∈ R. Hence W ≡ R is a subspace.
(iii) Let W be a subspace and W ≠ {0}. We prove that W = R. Clearly W ⊆ R. Let x ∈ R. Then for y ∈ W, y ≠ 0, we have x = (x/y).y. Since (x/y).y ∈ W, x ∈ W. This shows that R ⊆ W. Hence R = W.
7. Subspaces of R²: Subspaces of the vector space R2 over R are precisely {0}, R and lines passing through origin. That is,
(i) W≡{(0,0)} is a subspace of the vector space R2 over R.
(ii) W≡R2 is a subspace of the vector space R2 over R.
(iii) If W is a line passing through origin, then W is a subspace of the vector space R2 over R and
(iv) If W is a subspace of the vector space R2 over R, then W={0} or R2 or a line passing through origin.
Proof:
(i) It is similar to that of 5(i).
(ii) It is similar to that of 5(ii).
(iii) Clearly W ≡ {(x, y) ∈ R² : ax + by = 0}, for some a, b ∈ R, such that either a \ne 0 or b \ne 0. Notice that W is the line y = mx or x = 0, where m ∈ R. Clearly W ⊆ R² and (0 , 0) ∈ W. Let (r , s) , (t , u) ∈ W. Hence ar + bs = 0, at + bu = 0. Then (r , s) + (t , u) = (r+t , s+u) ∈ W, since a(r + t) + b(s + u)=[(ar + bs) + (at + bu)] = 0. Also α.(r, s) = (α.r , α.s) ∈W, since a(α.r) + b(α.s) = 0. Hence W is a subspace.
(iv) Let W be a subspace of R² other than {(0, 0)} and R². We prove that W is a line passing through the origin. Let (c, d) ∈ W such that (c, d) \ne 0. We claim that W is the line L passing through (c, d) and origin. Then L= {(x, y) ∈ R² : dx - cy = 0}. Clearly L ⊆ W and we prove that W ⊆ L. Let (m, n) ∈ W such that (m, n) ≠ (0, 0). To the contrary, let (m, n) ∉ L. For (x, y) ∈ R²; one can obtain α, β ∈ R such that (x, y)=α.(m, n) + β.(c, d). Then (x, y) ∈ W. Hence W ≡ R², a contradiction. This completes the proof.
8.Properties of Subspaces:
Let W₁ and W₂ be the subspaces of V. Then,
i. Intersection of W₁ and W₂ i.e., W₁ ∩ W₂ is also a subspace of V.
ii. Union of W₁ and W₂ i.e.,W₁ ⋃ W₂ may or may not be a subspace of V.
iii. Union of W₁ and W₂ i.e., W₁ ⋃ W₂ is a subspace of V if and only if W₁ ⊆ W₂ or W₂ ⊆ W₁.
iv. W₁ + W₂ ≡ {x + y : x ∈ W₁, y ∈ W₂} is a subspace of V.
v. W₁ + W₂ = span (W₁ ⋃ W₂).
vi. W₁ + W₂ is the smallest subspace containing W₁ and W₂.