Vector spaces, subspaces and linear independence

What a vector space is

Definition. A vector space over the real numbers is a set $V$ with two operations, vector addition and multiplication by a scalar, satisfying these rules for all $u, v, w \in V$ and scalars $a, b$:

1. Closure: $u + v \in V$ and $a u \in V$.
2. Addition is commutative and associative.
3. There is a zero vector: $v + 0 = v$.
4. Every $v$ has a negative: $v + (-v) = 0$.
5. $a(u + v) = au + av$, $(a + b)v = av + bv$, $a(bv) = (ab)v$, and $1v = v$.

Examples.

• $\mathbb{R}^n$, the column vectors with $n$ real entries.
• $P_n$, the polynomials of degree at most $n$, such as $3 - x + 2x^2$ in $P_2$.
• $M_{m \times n}$, the $m \times n$ real matrices.
• The continuous functions on $[0, 1]$.
• The solutions of a homogeneous system $Ax = 0$.

Non-examples.

• The polynomials of degree exactly 2. Adding $x^2$ and $-x^2 + x$ gives $x$, which has degree 1, so the set is not closed under addition.
• The vectors in $\mathbb{R}^2$ with $x \ge 0$. Multiplying by $-1$ leaves the set.
• The solutions of $Ax = b$ with $b \ne 0$. The zero vector is not a solution.

Subspaces

Definition. A subset $W$ of a vector space $V$ is a subspace if it is a vector space with the same operations.

Subspace test. It is enough to check three things:

1. $0 \in W$.
2. $u, v \in W \Rightarrow u + v \in W$.
3. $v \in W \Rightarrow a v \in W$ for every scalar $a$.

The last two combine into one check: $au + bv \in W$ for all $u, v \in W$ and all scalars $a, b$.

Exam point. A quick way to rule a set out is that it does not contain $0$. Every subspace of $\mathbb{R}^3$ is one of four kinds: $\{0\}$, a line through the origin, a plane through the origin, or $\mathbb{R}^3$ itself.

Solved examples

Example 1. Which of these are subspaces of $\mathbb{R}^3$?

• (a) $\{(x, y, z) : x + 2y - z = 0\}$. Yes. It is a plane through the origin, the null space of $[1\ 2\ {-1}]$.
• (b) $\{(x, y, z) : x + y = 1\}$. No. It does not contain 0.
• (c) $\{(x, y, z) : xy = 0\}$. No. It contains $(1, 0, 0)$ and $(0, 1, 0)$, but not their sum $(1, 1, 0)$. It is the union of two planes.
• (d) $\{(t, 2t, -t) : t \in \mathbb{R}\}$. Yes. It is a line through the origin.
• (e) $\{(x, y, z) : x \ge 0\}$. No. It is not closed under multiplying by $-1$.

Example 2. Is the set of $2 \times 2$ matrices with trace 0 a subspace of $M_{2 \times 2}$? Yes. The trace is linear: $\operatorname{tr}(aA + bB) = a\operatorname{tr}A + b\operatorname{tr}B = 0$. The set of invertible matrices is not a subspace: it does not contain the zero matrix, and $I + (-I) = 0$ is not invertible.

Unions and intersections.

• The intersection of two subspaces is always a subspace.
• A union of two subspaces is a subspace only if one contains the other.
• The sum $U + W = \{u + w : u \in U, w \in W\}$ is the smallest subspace containing both.

Linear combinations and span

A linear combination of $v_1, \dots, v_k$ is $c_1 v_1 + \dots + c_k v_k$. Their span is the set of all linear combinations, and it is always a subspace.

Example 3: is a vector in the span? Is $(3, 5, 7)$ in $\operatorname{span}\{(1, 1, 1), (1, 2, 3)\}$? We need $a(1, 1, 1) + b(1, 2, 3) = (3, 5, 7)$:

• $a + b = 3$;
• $a + 2b = 5$;
• $a + 3b = 7$.

The first two give $b = 2$ and $a = 1$. The third then gives $1 + 6 = 7$, which holds. Yes: $(3, 5, 7) = (1, 1, 1) + 2(1, 2, 3)$.

Matrix view. $b$ is in the span of the columns of $A$ exactly when $Ax = b$ has a solution, that is, when $\operatorname{rank}[A \mid b] = \operatorname{rank} A$.

Linear independence

Definition. Vectors $v_1, \dots, v_k$ are linearly independent if

$c_1 v_1 + \dots + c_k v_k = 0 \implies c_1 = \dots = c_k = 0 .$

Otherwise they are dependent, and at least one of them is a combination of the others.

Tests.

• Put the vectors as the columns of a matrix $A$. They are independent exactly when $\operatorname{rank} A = k$, that is, when $Ax = 0$ has only the zero solution.
• $n$ vectors in $\mathbb{R}^n$ are independent exactly when $\det A \ne 0$.
• More than $n$ vectors in $\mathbb{R}^n$ are always dependent.
• Any set containing the zero vector is dependent.
• Two vectors are dependent exactly when one is a multiple of the other.

Solved examples

Example 4. Are $(1, 2, 3)$, $(4, 5, 6)$ and $(7, 8, 9)$ independent?