# Vector Spaces and Subspaces

Physics I · Linear Algebra and Geometry · https://tryals.app/learn/physics-i/vector-spaces-and-subspaces

## Vector Spaces & Subspaces

A **vector space** $V$ over $\mathbb{R}$ is a set equipped with **addition** and **scalar multiplication** satisfying eight strict algebraic axioms.

To prove a subset $W \subseteq V$ is a **subspace**, you only need three checks. **Subspace test** criteria must all hold:

| Condition | Mathematical Form | Meaning |
| --- | --- | --- |
| **Zero vector** | $\vec{0} \in W$ | Must contain origin |
| **Addition** | $\vec{u} + \vec{v} \in W$ | Closed under add |
| **Scalar mult** | $c\vec{v} \in W$ | Closed under scaling |

**Common pitfall:** A subspace must contain the zero vector. The solutions of $Ax = b$ with $b \neq 0$ never form a subspace, however plane-like they look.

## Core Concepts & Subspaces

**Key concepts** build the structural anatomy of vector spaces:

| Concept | Definition | Function |
| --- | --- | --- |
| **Span** | All linear combinations | Generated space |
| **Independence** | No redundancy | Minimal vectors |
| **Basis** | Independent span | Coordinate system |
| **Dimension** | Basis vector count | Exact size |

For any matrix $A_{m \times n}$, its **fundamental subspaces** are the **Column space** $\mathrm{Col}(A) \subseteq \mathbb{R}^m$ and **Null space** $\mathrm{Null}(A) \subseteq \mathbb{R}^n$.

**Rank-nullity:** $\mathrm{rk}(A) + \mathrm{null}(A) = n$. **Tip:** To find a basis for $\mathrm{Col}(A)$, row-reduce and take pivot columns.

## Practice questions

4 of this lesson's 13 practice questions, with answers. The full set is in the app.

### 1. The set of vectors with $x + y + z = 0$ is a subspace, but $x + y + z = 1$ is not. Why does the second set fail to be a subspace?

A. It is unbounded, which subspaces are not permitted to be
B. It is not a plane but a curved surface
C. It lacks the zero vector and is not closed under addition
D. It has three variables rather than two

**Answer:** C. It lacks the zero vector and is not closed under addition

**Why:** A subspace must pass through the origin and stay closed. The second set is an affine plane — the same shape, moved — which is precisely why the solutions of $Ax = b$ are not a subspace while those of $Ax = 0$ are.

Page: https://tryals.app/practice/physics-i/vector-spaces-and-subspaces/the-set-of-vectors-with-x-y-z-0-is-a-subspace-but-x-y-z

### 2. Order these steps to check if a set S is a subspace of ℝⁿ.

**Answer:**

1. Verify the zero vector is in S
2. Check closure under addition: u,v $\in S$ ⟹ u+v $\in S$
3. Check closure under scalar multiplication: u $\in S, c$ ∈ ℝ ⟹ cu $\in S$
4. Conclude S is a subspace

**Why:** All three conditions must hold: contains zero, closed under addition and scaling.

Page: https://tryals.app/practice/physics-i/vector-spaces-and-subspaces/order-these-steps-to-check-if-a-set-s-is-a-subspace-of

### 3. Do the solutions of $x + y + z = 0$ form a **subspace** of $\mathbb{R}^{3}$? And those of $x + y + z = 6$?

A. Both are subspaces: each is a plane
B. First yes, second no
C. Neither: subspaces must be lines
D. First no, second yes

**Answer:** B. First yes, second no

**Why:** The plane through the origin passes every test; the shifted plane fails immediately ($\mathbf{0}$ missing, sums drift to 12). Subspaces must pass through the origin, homogeneous equations ($= 0$) build subspaces, inhomogeneous ones ($= c$) build offset copies that are not.

Page: https://tryals.app/practice/physics-i/vector-spaces-and-subspaces/do-the-solutions-of-x-y-z-0-form-a-subspace-of-and-those-of

### 4. Sort each set of vectors in $\mathbb{R}^{3}$: linearly **independent** or **dependent**?

**Answer:**

- Independent: $(1,0,0),\ (0,1,0),\ (0,0,1)$, $(1,1,0),\ (0,1,1)$
- Dependent: $(1,2,3),\ (2,4,6)$, Any four vectors in $\mathbb{R}^{3}$

**Why:** Independence = no redundancy = each vector opens a genuinely new direction. Three dimensions can host at most three independent directions, so four vectors are automatically dependent, a counting law with deep consequences (rank, basis, dimension).

Page: https://tryals.app/practice/physics-i/vector-spaces-and-subspaces/sort-each-set-of-vectors-in-linearly-independent-or-dependent