Vector Spaces & Subspaces

You already know how to add arrows and stretch them — that’s the whole of working inside a vector space. A vector space is just the polite name for any setting where addition and scaling behave the way you’d expect, and the plane R² and 3D space R³ are the everyday examples. The interesting question — and the one GATE keeps asking — is which subsets of a vector space are themselves vector spaces. These are called subspaces, and they pass or fail on three quick checks. The exam loves sets that pass two of the three and quietly miss the third. (Subspaces aren’t just exam fodder: the low-dimensional subspace that best captures your data is exactly what PCA hunts for, and a model’s column space is the set of outputs it can ever produce.)

The subspace test — three conditions

A subset W of a vector space is a subspace if and only if all three hold:

1. Contains the zero vector — the origin 0 must be in W.
2. Closed under addition — if u and v are in W, then u + v is in W.
3. Closed under scalar multiplication — if v is in W and c is any scalar, then cv is in W.

Geometrically in R³ the only subspaces are: the origin alone, every line through the origin, every plane through the origin, and all of R³. Anything that misses the origin, bends, or is bounded fails.

The fastest screen is condition 1: if the origin is not in the set, stop — it is not a subspace. That single check kills every “offset” set instantly.

Drag the two arrows and watch what their sum and scalings do. As long as both arrows live in your candidate subset, every sum and every scaled copy must too — or the closure conditions break. Try lining the arrows up along a line through the origin and notice that addition and scaling never push you off that line.

How GATE asks this

The pattern is a clean MCQ or MSQ: a list of subsets of R² or R³ is given, and you select which ones are subspaces. GATE DA 2024 ran exactly this format. The designed trap mixes three flavours of set: a genuine subspace (a linear equation through the origin), an affine set (a linear equation equal to a nonzero constant, so it misses the origin), and a set defined with squared coordinates (which looks algebraic but breaks closure under scaling). Test all three conditions, but lead with the zero-vector check — it resolves most options on its own.

Worked example — three sets, one test each

Decide which of these subsets of R² are subspaces: (a) {(x, y) : x + y = 0}, (b) {(x, y) : x² + y² ≤ 1}, (c) {(x, y) : x + y = 1}.

Set (a) — x + y = 0. Subspace. The origin (0, 0) satisfies 0 + 0 = 0, so condition 1 holds. If (x₁, y₁) and (x₂, y₂) both satisfy it, their sum gives (x₁ + x₂) + (y₁ + y₂) = 0, so it is closed under addition. Scaling by c gives cx + cy = c(x + y) = c·0 = 0 — closed. All three pass; this is the line through the origin with slope -1.

Set (b) — x² + y² ≤ 1 (the unit disk). NOT a subspace. The origin is inside, so condition 1 passes — but closure fails. Take (1, 0), which is in the disk, and scale by c = 2: you get (2, 0), and 2² + 0² = 4 > 1, which is outside. Closure under scalar multiplication is broken, so it is not a subspace. (The squared terms are the giveaway — they make the set bounded.)

Set (c) — x + y = 1 (affine). NOT a subspace. Check condition 1 first: (0, 0) gives 0 + 0 = 0 ≠ 1, so the origin is not in the set. It fails immediately — this is the line x + y = 0 shifted off the origin.

So only (a) is a subspace. Three sets, three different reasons to keep or reject.

Quick check