Vector spaces formulas

Vector spaces, subspaces, and linear dependence and independence. 8 formulas for GATE DA Linear Algebra, each with when to use it. Each is explained in the topic notes.

All 8 Vector spaces formulas

SUBSPACE TEST

W \subseteq V is a subspace ⟺ 0 \in W and u, v \in W, c \in R \Rightarrow u + v \in W and c u \in W

Applies to any subset W of a vector space V. If the zero vector is missing, W fails immediately.

SUBSPACES UNDER INTERSECTION AND SUM

dim (W_{1} + W_{2}) = dim W_{1} + dim W_{2} - dim (W_{1} \cap W_{2})

W1 and W2 are finite-dimensional subspaces of a vector space V. The intersection and the sum W1+W2 (all u+v with u in W1, v in W2) are always subspaces. The union is a subspace exactly when one of the two contains the other.

SPAN

span {v_{1}, \dots, v_{k}} = {c_{1} v_{1} + \dots + c_{k} v_{k} : c_{i} \in R}

The span is always a subspace, and it is the smallest subspace containing the given vectors.

MEMBERSHIP IN A COLUMN SPACE

b \in span {columns of A} ⟺ A x = b is consistent ⟺ rank [A ∣ b] = rank A

A is m x n real, b is in R^m. Use it to decide whether a vector lies in the span of given vectors.

DEFINITION OF LINEAR INDEPENDENCE

c_{1} v_{1} + \dots + c_{k} v_{k} = 0 \Rightarrow c_{1} = \dots = c_{k} = 0

The vectors are independent if only the trivial combination gives the zero vector; otherwise they are dependent. For k >= 2 this is equivalent to: no vector in the set is a combination of the others. Also equivalent: the list v1, ..., vk is dependent exactly when v1 = 0 or some vj is a combination of v1, ..., v(j-1).

BASIS FACTS

dim V = n \Rightarrow any n independent vectors, or any n spanning vectors, form a basis

V is finite-dimensional. Any independent set has at most n vectors, any spanning set has at least n, and an independent set can be extended to a basis. If W is a subspace of V, then dim W <= n, with equality only when W = V.

DIMENSIONS OF THE MATRIX SUBSPACES

dim Col (A) = dim Row (A) = r, dim Null (A) = n - r, dim Null (A^{T}) = m - r

A is real m x n of rank r. Row rank equals column rank, so rank(A) = rank(A^T), and r <= min(m, n). The middle result, rank plus nullity equals the number of columns n, is the rank-nullity theorem.

RANK-NULLITY THEOREM

rank (A) + nullity (A) = n

n is the number of columns of A. Nullity is the dimension of the null space.

Where this was asked: 2024 (1), 2025 (1), 2026 (1). See the 3 previous-year questions on Vector spaces. This shows what was asked, not what will be asked.

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Vector spaces: the notes and 3 solved previous-year questions