Download Contents Definition of a Subspace of a Vector Space

Math1300:MainPage/Subspaces
Contents
• 1 Definition of a Subspace of a Vector Space
♦ 1.1 Theorem (Subspace Test)
♦ 1.2 Corollary (A Subspace must contain 0)
• 2 Examples of Subspaces and the Subspace Test
♦ 2.1 Polynomials
◊ 2.1.1 W = Pm is a subspace of V = Pn
if
◊ 2.1.2 and V = Pn
♦ 2.2 Euclidean n-space
◊ 2.2.1 A line through in 2-space
◊ 2.2.2 A line through in 3-space
◊ 2.2.3 A plane through in 3-space
◊ 2.2.4 The set of solutions to Ax=0
♦ 2.3 Mnn, the n by n matrices
◊ 2.3.1 Upper Triangular, Lower
Triangular, Diagonal Matrices
◊ 2.3.2 Symmetric Matrices
◊ 2.3.3 Matrices with Trace 0
◊ 2.3.4 Solutions to Ax=0
• 3 Some Subsets that are not Subspaces
♦ 3.1 Polynomials
◊ 3.1.1 Polynomials p(x) such that p(0)
>0
◊ 3.1.2 Polynomials with
♦ 3.2 Euclidean n-space
◊ 3.2.1 The First Quadrant
◊ 3.2.2 Axes in n-space
Definition of a Subspace of a Vector Space
By definition a vector space consists of a set V of vectors with addition and scalar multiplication defined so as to
satisfy ten properties: five involve addition A1-A5 and five involve scalar multiplication M1-M5. A subspace of a
vector space is a set W which is a subset of V which, using the same addition and scalar multiplication as in V, is a
vector space in its own right.
A hint at the structure was given here when we considered 2-space and 3-space. We saw it was reasonable to
define a 0-space to be just the set
A 1-space was a line containing
A 2-space could be obtained by
adding a new axis to the 1-space, a 3-space could be obtained from a 2-space by adding another axis, etc. From
this perspective, the 0-space is contained in a 1-space which is in turn contained in a 2-space and then in a
3-space. This indicates that a k-space is a subspace of a k + 1-space for k = 0,1,2,3.
For a more specific example, consider the vector space Pn, all polynomials of degree n or less. Clearly Pn is
contained in Pn + 1 for any
and so Pn is a subspace of Pn + 1.
Suppose we have a set W that is a subset of a vector space V. Is it a subspace? On the face of it, we have to see if
it satisfies the five additive properties A1-A5 and five multiplicative properties M1-M5. But, in fact, the next
theorem shows that we need only check A1 and M1.
Contents
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Theorem (Subspace Test)
Suppose W is a nonempty subset of a vector space V. Then W is a subspace if, for any vectors
scalar r,
in W and any
A1:
M1:
is in W.
is in W.
(W is closed under addition.)
(W is closed under scalar multiplication.)
Proof We have to show that the remaining eight properties A2-A5 and M2-M5 are valid. Of these, A2, A5, M2,
M3, M4 and M5 are valid for the same reason. Each is an equation that must be valid for vectors in W. But W is
contained in V, and each equation is already valid for all vectors in V. So the vectors in W inherit the properties
from V.
This only leaves A3 and A4. If
For A3, we need to show that
by A1,
is in W., then, by M1,
in in W. If
is in W and so A4 is valid.
is any vector in W, then we have just seen that
is in W and,
is in W. Hence A3 is valid.
Corollary (A Subspace must contain 0)
If W is a subset of a vector space V, and
is not in W, then W is not a subspace.
Examples of Subspaces and the Subspace Test
To show that a subset W of a vector space is a subspace we must show two things:
1. (A1) The subset is closed under addition, that is,
and
2. (M1) The subset is closed under scalar multiplication, that is,
implies
and r a scalar implies
Polynomials
W = Pm is a subspace of V = Pn if
Closure under addition: if p(x) with degree s and q(x) with degree t are in Pm, then
degree of (p + q)(x) is then at most the maximum of s and t, and hence at most m.
and
Closure under scalar multiplication: The degree of (rp)(x) is the same as the degree of p(x) if
then
Hence the degree is at most m and W is a subspace.
The
If r = 0,
and V = P
n
Closure under addition: If p(x) and q(x) are in W, then p(0) = q(0) = 0. Hence (p + q)(0) = p(0) + q(0) = 0 + 0 = 0.
and p + q is in W.
Closure under scalar multiplication: If p(x) is in W, then p(0) = 0. Hence (rp)(0) = rp(0) = r0 = 0, and rp is in W.
Theorem (Subspace Test)
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Math1300:MainPage/Subspaces
Euclidean n-space
A line through
in 2-space
Let the equation of the line be y = mx. Two vectors
only if y1 = mx1 and y2 = mx2.
Closure under addition:
mx2 = m(x1 + x2). Hence
and
and y + y = mx +
1
is also on the line.
Closure under scalar multiplication:
is also on the line.
A line through
are on the line if and
2
1
Since ry = rmx = mrx , the vector
1
1
1
in 3-space
A line in 3-space through (x1,y1z1) has an equation of the form (x,y,z) = (x1,y1z1) + t(a,b,c) for
If the line passes through
then the line consists of points (x,y,z) satisfying (x,y,z) = t(a,b,c). Hence our subset W
is the the set of (x,y,z) satisfying (x,y,z) = t(a,b,c).
Closure under addition: if
Thus
in W.
and
and
is
and so
Closure under scalar multiplication: if
and so
A plane through
are in W, then
is in W, then
is in W.
in 3-space
The point-normal equation of a plane through
satisfying that equation.
is ax + by + cz = 0, so W is the set of vectors
Closure under addition: If
and
are in W, then ax + by + cz = 0 and
1
1
1
ax + by + cz = 0.
and
2 + y ) + c(z + z ) = ax + by + cz + ax + by + cz = 0 + 0 = 0.
a(x21 + x22) + b(y
1
2
1
2
1
1
1
2
2
2
Closure under scalar multiplication:
+ by1 + cz1) = r0 = 0.
and arx + bry + crz = r(ax
1
1
1
1
The set of solutions to Ax=0
Let A be a given
If
and
matrix and let
are in W, then
and
Notice that W is a subset of
Hence
and so
Euclidean n-space
is in W.
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Similarly
and
is in W.
This subspace is called the null space of the matrix A.
Mnn, the n by n matrices
Upper Triangular, Lower Triangular, Diagonal Matrices
Matrices A and B are upper triangular if ai,j = bi,j = 0 whenever i > j. In this case, ai,j + bi,j = 0 + 0 = 0, and so A +
B is also upper triangular. In addition, in this case rai,j = r0 = 0, and so rA is also upper trianglar. Hence the set of
upper triangular matrices Is closed under addition and scalar multiplication, and so is a subspace.
For lower triangular matrices and diagonal matrices, the condition on the subscripts changes to i < j and
respectively. The remaining arguments are unchanged.
Symmetric Matrices
Matrices A and B are symmetric if A = AT and B = BT. Then (A + B)T = AT + BT = A + B so A + B is also
symmetric. In addition, (rA)T = rAT = rA and so rA is symmetric. Hence the set of symmetric matrices is closed
under addition and scalar multiplication.
Matrices with Trace 0
Recall that the trace of a matrix is the sum of the diagonal elements. Matrices A and B have trace 0 if
and
Then the trace of A + B is
Similarly for the trace of rA:
Hence the set of matrices with trace 0 is closed under addition and scalar multiplication and is a subspace.
Solutions to Ax=0
Suppose that A is a given
If
and
are in W, then
matrix. Let W be the set of vectors
such that
Notice that
is in
Hence
and
The set of solutions to Ax=0
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This means
of the matrix A.
and
are both in W, and so W is a subspace. This subspace of
is called the null space
Some Subsets that are not Subspaces
A subset is not a subspace if either of the following two conditions does not hold:
1. A1: The subset is closed under addition, that is,
and
in W implies
2. M1: The subset is closed under scalar multiplication, that is,
in W implies
in W.
in W for any scalar r.
Polynomials
Polynomials p(x) such that p(0) > 0
Let
If polynomials p(x) and q(x) are in W, that is, they satisfy p(0) > 0 and q(0) >
0, then (p + q)(0) = p(0) + q(0) > 0 and so the set of polynomials W is closed under addition.
However, for any polynomial p(x) in W, let r = − 1. Then (rp)(0) = ( − 1)p(0) < 0 and so rp(x) is not in W. Since
M1 is not always satisfied, the subset is not a subspace.
Polynomials with
Let W be the set of all polynomials in P2: p(x) = a2x2 + a1x + a0 satisfying
scalar multiplication since rp(x) = ra2x2 + ra1x + ra0, and
implies
This set is closed under
W is not closed under addition: if p(x) = x2 + 1 and q(x) = x2 − 1, then both p(x) and q(x) are in W. However p(x) +
q(x) = x2 + 1 + x2 − 1 = 2x2 and 2x2 is not in W.
Notice that P2 can be replaced by Pn for any n > 0 and the same result it true with the identical proof.
Euclidean n-space
The First Quadrant
Let W be the vectors (x,y) in 2-space with
and
represent addition and scalar multiplication geometrically:
Solutions to Ax=0
This set is called the first quadrant. We can
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Clearly, the sum of two vectors in the first quadrant remains in the first quadrant; the same hold true when a
vector in the first quadrant is multiplied by a positive constant. But it is not true when the vector in multiplied by a
negative constant. Hence the first quadrant is closed under addition but not under scalar multiplication.
The geometry makes it clear that there is nothing special about the first quadrant. Any of the four quadrants may
be used.
Axes in n-space
The vectors in Eulcidean n-space are of the form
The first axis L is the set of vectors of the
1
form
the second axis L is the set of vectors of the form
and the other axes L ,
2
k
have the analogous definition: L is the set of vectors
with x = 0 if
k
i
Let W be the set of all vectors on the axes. An alternative description of W is the set of vectors with at least n − 1
coordinates equal to zero.
W is closed under scalar multiplication since any coordinate equal to zero remains so after scalar multiplication.
But it is not closed under addition. Indeed, the sum of any two nonzero vectors on different axes will not be in W.
The First Quadrant
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