13_Operations_on

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Operations on Vector Spaces¶

Union, Intersection¶

Let $W,W' \le V$.

Is $W \cup W'$ necessaily a subspace of $V$?
Is $W \cap W'$ necessarily a subspace of $V$?

Proposition 1. The intersection of two subspaces is a subspace.

Proof. Suppose $W_1 W_2 \le V$. Then $\vz$ is in both $W_1$ and $W_2$ and hence in $W_1 \cap W_2$. Take any $c \in K$ and $\vw, \vw' \in W_1 \cap W_2$. Then by the Proposition 1 in Note 12, $c\vw + \vw'$ is in both $W_1$ and $W_2$ and hence their intersection.

Sum of vector spaces¶

Let $X, Y \le V$, the sum of $X$ and $Y$, denoted by $X +Y$, is defined to be the set $$ \{ \vx + \vy \colon \vx \in X, \vy \in Y\} $$ equipped with the operations of $V$. Show that $X +Y \le V$.

The sum $X+Y$ is a direct sum if $X \cap Y = \{\vz_V\}$. In that case, we denote the sum by $X \oplus Y$.

What is $X + Y$ when $X$, $Y$ are the following subspaces of $\Rr^3$?

a. $X = xz$-plane, $Y = yz$-plane.

b. $X = xz$-plane, $Y =y$-axis.

c. $X = xz$-plane, $Y = z$-axis.

Which of these sums are direct?

a. $X+Y = \Rr^3$ but the sum is not direct since $X \cap Y = z$-axis.

b. $X+Y = \Rr^3$ and the sum is direct since $X \cap Y =\{\sp{0,0,0} \}$.

c. $X+Y = xz$-plane and the sum is not direct sincee $X \cap Y = z$-axis.

Direct Product¶

Let $X, Y$ be vector spaces. The set $X \times Y$ equipped with the operations:

$(\vx_1,\vy_1) + (\vx_2, \vy_2) = (\vx_1 +_{X} \vx_2, \vy_1 +_{Y} \vy_2)$
$\lambda(\vx , \vy) = (\lambda \cdot_{X} \vx, \lambda \cdot_{Y} \vy)$

is a vector space.

It is the direct product of $X$ and $Y$. The zero vector of $X \times Y$ is $\vz:=(\vz_X, \vz_Y)$.

Example. $K^n \times K^m = K^{n+m}$.

Quotients¶

Suppose $W \le V$. Define a relation $\sim_W$ on $V$ by $\vv \sim_W \vv'$ if $\vv-\vv' \in W$.

Check that $\sim_W$ is an equivalence relation.

Let $V/W = \{[\vv]_W \colon \vv \in V\}$ be the set of equivalence classes $\sim_W$.

Check that the operations $[\vv]_W + [\vv']_W = [\vv+\vv']_W$ and $\lambda [\vv]_W = [\lambda \vv]_W$ are well-defined.

$V/W$ equipped with these operations is a vector space called the quotient (space) of $V$ by $W$.

Take $V = \Rr^3$ and $W = z$-axis.

Describe the equivalent class of $\sp{3,3,1}$ with respect to $\sim_W$.
Find $[\sp{3,3,1}] + [\sp{4,6,0}]$ and $2[\sp{2,8,1}]$ in $V/W$.

$[\sp{3,3,1}] = \{\sp{3,3,\lambda} \colon \lambda \in \Rr\}$.
$[\sp{3,3,1}] + [\sp{4,6,0}] = [\sp{7,9,1}]$ and $2[\sp{2,8,1}] = [\sp{4,16,2}]$