Linear Algebra, 2020a

102 Chapter Two. Vector Spaces
unrestricted linear combinations of the two {c 1 (3x − x 2 )+c 2 (2x) | c 1 ,c 2 ∈ R}.
Clearly polynomials in this span must have a constant term of zero. Is that
necessary condition also sufficient?
We are asking: for which members a 2 x 2 + a 1 x + a 0 of P 2 are there c 1 and c 2
such that a 2 x 2 + a 1 x + a 0 = c 1 (3x − x 2 )+c 2 (2x)? Polynomials are equal when
their coefficients are equal so we want conditions on a 2 , a 1 , and a 0 making that
triple a solution of this system.
−c 1 = a 2
3c 1 + 2c 2 = a 1
0 = a 0
Gauss’s Method and back-substitution gives c 1 =−a 2 , and c 2 =(3/2)a 2 +
(1/2)a 1 , and 0 = a 0 . Thus as long as there is no constant term a 0 = 0 we can
give coefficients c 1 and c 2 to describe that polynomial as an element of the
span. For instance, for the polynomial 0 − 4x + 3x 2 , the coefficients c 1 =−3 and
c 2 = 5/2 will do. So the span of the given set is [S] ={a 1 x + a 2 x 2 | a 1 ,a 2 ∈ R}.
Incidentally, this shows that the set {x, x 2 } spans the same subspace. A space
can have more than one spanning set. Two other sets spanning this subspace
are {x, x 2 , −x + 2x 2 } and {x, x + x 2 ,x+ 2x 2 ,...}.
2.19 Example The picture below shows the subspaces of R 3 that we now know
of: the trivial subspace, lines through the origin, planes through the origin, and
the whole space. (Of course, the picture shows only a few of the infinitely many
cases. Line segments connect subsets with their supersets.) In the next section
we will prove that R 3 has no other kind of subspace, so in fact this lists them all.
This describes each subspace as the span of a set with a minimal number
of members. With this, the subspaces fall naturally into levels — planes on one
level, lines on another, etc.
⎛
⎞
⎛
⎞
{x ⎝ 1 0⎠ + y ⎝ 0 1⎠}
0 0
⎛
✄
✄
⎞
{x ⎝ 1 0⎠}
0
✘✘✘✘✘✘✘
✘✘⎛
✘ ⎞ ⎛ ⎞ ✏ ✏✏✏✏ ⎛ ⎞
{x ⎝ 1 0⎠ + z ⎝ 0 0⎠}
0 1
⎛
{x ⎝ 1 ⎞ ⎛
0⎠ + y ⎝ 0 ⎞ ⎛
1⎠ + z ⎝ 0 ⎞
0⎠}
0 0 1
⎛
{x ⎝ 1 1⎠ + z ⎝ 0 0
0 1
❆
❍
✏
✏ ✏✏✏ ❍❍❍ ❆
⎛
{y ⎝ 0 ⎞ ⎛
1⎠}
{y ⎝ 2 ⎞ ⎛
1⎠}
{y ⎝ 1 ⎞
1⎠} ···
❳ 0
0
1
❳ ❳
❳❳ ❍
❳ ❳❳ ❍❍❍
❳
❆ ⎛ ⎞
❳❳
❳
{ ⎝ 0 0⎠}
0
⎞
⎠} ···