Quotient Spaces, linear algebra II lesson

A sensor records the position of a point in space but ignores its height. It cannot tell $(2, 3, 0)$ from $(2, 3, 7)$ , or from any other point on the vertical line through them. Is there a sensible way to treat each such line as one object, and then to add and scale these objects? The quotient space does exactly this, and the real work is checking that every answer is the same whichever point of a line we happen to pick.

Key terms

Equivalence modulo a subspace

Definition Let $V$ be a vector space over $\mathbb{F}$ , where $\mathbb{F}$ is $\mathbb{R}$ or $\mathbb{C}$ , and let $W$ be a subspace of $V$ . Two vectors $v, u \in V$ are equivalent modulo $W$ , written $v \sim u$ , when

v - u \in W.

This is an equivalence relation. It is reflexive because $v - v = 0 \in W$ , symmetric because $u - v = -(v - u)$ and $W$ is closed under negatives, and transitive because $v - z = (v - u) + (u - z)$ and $W$ is closed under addition. No finite dimension and no inner product are needed.

In plain words Two vectors count as the same when they differ only by a vector of $W$ , the directions we have agreed to ignore.

Example In $V = \mathbb{R}^3$ let $W$ be the vertical axis, spanned by $e_3 = (0, 0, 1)$ . The difference $(2, 3, 7) - (2, 3, 0) = (0, 0, 7)$ lies in $W$ , so $(2, 3, 7) \sim (2, 3, 0)$ . In general two vectors are equivalent exactly when their first two coordinates agree. The subspace hypothesis matters. If the set $\{1\}$ in $\mathbb{R}$ were used in place of a subspace, the condition $v - u \in \{1\}$ would not even be reflexive, since $v - v = 0$ is not in the set.

Coset

Definition For $v \in V$ , the coset of $v$ modulo $W$ is the set

v + W = \{\, v + w \mid w \in W \,\}.

It is the equivalence class of $v$ , and $v$ is called a representative of it. Two cosets satisfy $v + W = u + W$ exactly when $v - u \in W$ . If $v = u + w_0$ with $w_0 \in W$ , then $v + w = u + (w_0 + w)$ , and $w_0 + w$ runs over all of $W$ as $w$ does, so the two sets agree. Conversely, if the cosets are equal, then $v = v + 0$ lies in $u + W$ , so $v = u + w$ for some $w \in W$ and $v - u \in W$ . Two cosets are either equal or disjoint, because a common point $v + w_1 = u + w_2$ gives $v - u = w_2 - w_1 \in W$ . So the cosets split $V$ into pieces that do not overlap. A coset is a subspace only when it contains $0$ , which happens exactly when $v \in W$ and the coset is $W$ itself.

In plain words A coset is a copy of $W$ shifted so that it passes through $v$ . Every point on it is an equally good name for it.

Example With $W$ the vertical axis in $\mathbb{R}^3$ , the coset $(2, 3, 0) + W$ is the vertical line $\{(2, 3, c) \mid c \in \mathbb{R}\}$ . It equals $(2, 3, 7) + W$ , and it is not a subspace, since it misses the origin. The coset of any vector on the axis, such as $(0, 0, 5) + W$ , is the axis itself.

Quotient space

Definition The quotient space $V/W$ is the set of all cosets $v + W$ with $v \in V$ , with the operations

\begin{aligned} (v + W) + (u + W) &= (v + u) + W, \\[0.4em] a(v + W) &= av + W \end{aligned}

for $v, u \in V$ and $a \in \mathbb{F}$ . These rules are well defined, as the next key term shows, and with them $V/W$ is a vector space over the same field $\mathbb{F}$ . Its zero vector is the coset $0 + W = W$ , and the negative of $v + W$ is $-v + W$ . Every other axiom follows by doing the matching calculation on representatives and taking the coset of the result.

In plain words The elements of $V/W$ are whole cosets, not vectors of $V$ . To add two cosets, add any representatives and take the coset of the sum.

Example In $\mathbb{R}^3$ modulo the vertical axis,

\begin{aligned} &\big((2, 3, 0) + W\big) + \big((1, -1, 0) + W\big) \\[0.4em] &\quad = (3, 2, 0) + W. \end{aligned}

Choosing the representatives at heights $7$ and $-4$ instead gives $(2, 3, 7) + (1, -1, -4) = (3, 2, 3)$ . That is a different vector, but it lies on the same vertical line, so the resulting coset is the same. Likewise $2\big((2, 3, 7) + W\big) = (4, 6, 14) + W$ , which is the coset $(4, 6, 0) + W$ .

Well-defined operation

Definition A rule on $V/W$ is usually written with a representative, so it is a genuine function only if equivalent representatives always give the same result. For a function $f$ on $V$ , the rule $v + W \mapsto f(v)$ is well defined exactly when, for all $v, u \in V$ ,

v - u \in W \implies f(v) = f(u).

For a rule with several inputs, every input may be replaced by an equivalent representative. Coset addition and scaling pass this test. If $v' = v + w_1$ and $u' = u + w_2$ with $w_1, w_2 \in W$ , then

\begin{aligned} (v' + u') - (v + u) &= w_1 + w_2 \in W, \\[0.4em] av' - av &= aw_1 \in W, \end{aligned}

so the sum and the scalar multiple land in the same cosets whichever representatives we use. Both checks use only that $W$ is a subspace.

In plain words A formula written with one representative must give an answer that all the other representatives agree with. Trying one or two choices is evidence, not proof, so the check has to cover every choice at once.

Example In $\mathbb{R}^3$ modulo the vertical axis, the rule $(a, b, c) + W \mapsto a + b$ is well defined, because moving along a vertical line changes only $c$ . The rule $(a, b, c) + W \mapsto c$ is not. The vectors $(0, 0, 0)$ and $(0, 0, 1)$ name the same coset $W$ , yet the rule would assign it both $0$ and $1$ .

Quotient basis

Definition Let $V$ be finite-dimensional over $\mathbb{F}$ and let $W$ be a subspace. Take a basis $w_1, \dots, w_r$ of $W$ and extend it to a basis $w_1, \dots, w_r, v_1, \dots, v_s$ of $V$ . Then the cosets $v_1 + W, \dots, v_s + W$ form a basis of $V/W$ , called a quotient basis. They span, because any $x = \sum_i c_i w_i + \sum_j d_j v_j$ satisfies

x + W = \sum_{j=1}^{s} d_j (v_j + W),

the terms with $w_i$ falling into the zero coset. They are independent, because $\sum_j d_j (v_j + W) = W$ means $\sum_j d_j v_j \in W$ , so $\sum_j d_j v_j = \sum_i c_i w_i$ for some scalars $c_i$ , and the independence of the extended basis forces every $d_j = 0$ . When $W \ne V$ the extension is not unique, and different extensions can give different quotient bases.

In plain words The directions of $W$ become zero in the quotient. Only the added directions survive, and among them nothing is lost or repeated.

Example For $V = \mathbb{R}^3$ and $W = \operatorname{span}\{e_3\}$ , extend $e_3$ by $e_1$ and $e_2$ . The cosets $e_1 + W$ and $e_2 + W$ form a quotient basis, and

\begin{aligned} (a, b, c) + W &= a(e_1 + W) \\[0.4em] &\quad + b(e_2 + W). \end{aligned}

Extending by $(1, 0, 5)$ and $(0, 1, -2)$ instead gives different representatives but the same cosets, since $(1, 0, 5) + W = e_1 + W$ and $(0, 1, -2) + W = e_2 + W$ .

Dimension of a quotient space

Definition If $V$ is finite-dimensional over $\mathbb{F}$ and $W$ is a subspace of $V$ , then

\dim (V/W) = \dim V - \dim W.

This follows by counting the quotient basis above, which has $s = (r + s) - r$ members. In particular $V/W$ is the zero space exactly when $W = V$ , and $V/\{0\}$ has the same dimension as $V$ .

In plain words Each dimension of $W$ takes one dimension away from the quotient. Which directions survive depends on $W$ , so a quotient is not found by deleting an arbitrary coordinate.

Example In $V = \mathbb{R}^2$ let $W = \operatorname{span}\{(1, 1)\}$ , so $\dim(V/W) = 2 - 1 = 1$ . Since

(x, y) - (x - y)\,e_1 = (y, y) \in W,

every coset satisfies $(x, y) + W = (x - y)(e_1 + W)$ . The coset $e_1 + W$ is not zero because $e_1 \notin W$ , so it alone is a quotient basis. The cosets of $(3, 1)$ and $(5, 3)$ both have coordinate $2$ , and indeed they are equal, since $(5, 3) - (3, 1) = (2, 2) \in W$ . The vector $(1, 1)$ could not represent a basis element, because its coset is the zero coset.

Check yourself

In $\mathbb{R}^3$ let $W = \operatorname{span}\{(1, 2, -1)\}$ . Which of the cosets $(2, -3, 2) + W$ and $(3, 3, 0) + W$ equals $(4, 1, 0) + W$ ?

Show the answer

Compare differences of representatives. The difference $(4, 1, 0) - (2, -3, 2) = (2, 4, -2) = 2(1, 2, -1)$ lies in $W$ , so $(2, -3, 2) + W = (4, 1, 0) + W$ . The difference $(4, 1, 0) - (3, 3, 0) = (1, -2, 0)$ is not a multiple of $(1, 2, -1)$ , so $(3, 3, 0) + W$ is a different coset.

In $\mathbb{R}^2$ let $W = \operatorname{span}\{(1, 1)\}$ . Decide whether each rule is well defined on $\mathbb{R}^2/W$ , first $(x, y) + W \mapsto x$ and then $(x, y) + W \mapsto x - y$ .

Show the answer

The first rule is not well defined. The vectors $(0, 0)$ and $(1, 1)$ represent the same coset $W$ , but the rule gives $0$ for one and $1$ for the other. The second rule is well defined. Any two representatives of one coset have the form $(x, y)$ and $(x + c, y + c)$ with $c \in \mathbb{R}$ , and $(x + c) - (y + c) = x - y$ . A single pair that happens to agree would not prove this, which is why the argument works with an arbitrary $c$ .

In $\mathbb{R}^4$ let $W = \operatorname{span}\{(1, 1, 0, 0), (0, 0, 1, 1)\}$ . Find $\dim(\mathbb{R}^4/W)$ , a quotient basis, and the coordinates of $(5, 2, 1, 4) + W$ in that basis.

Show the answer

The two spanning vectors are independent, so $\dim W = 2$ and $\dim(\mathbb{R}^4/W) = 4 - 2 = 2$ . The matrix with columns $(1, 1, 0, 0)$ , $(0, 0, 1, 1)$ , $e_1$ and $e_3$ has determinant $-1$ , so these four vectors form a basis of $\mathbb{R}^4$ , and $e_1 + W$ , $e_3 + W$ is a quotient basis. For any vector,

\begin{aligned} &(x_1, x_2, x_3, x_4) - (x_1 - x_2)e_1 \\[0.4em] &\quad - (x_3 - x_4)e_3 \\[0.4em] &\quad = (x_2, x_2, x_4, x_4) \in W, \end{aligned}

so the coordinates of the coset are $x_1 - x_2$ and $x_3 - x_4$ . For $(5, 2, 1, 4)$ they are $3$ and $-3$ . As a check,

\begin{aligned} &(5, 2, 1, 4) - 3e_1 + 3e_3 \\[0.4em] &\quad = 2(1, 1, 0, 0) + 4(0, 0, 1, 1). \end{aligned}

Quotient Spaces

Key terms

Equivalence modulo a subspace

Coset

Quotient space

Well-defined operation

Quotient basis

Dimension of a quotient space

Check yourself

How this lesson connects

Builds on

Used later in