Vector Algebra
Real Euclidean Vector Spaces
The idea of a vector space can be extended to include objects that you would not initially consider to be ordinary vectors. Matrix spaces. Consider the set M2x3( R) of 2 by 3 matrices with real entries. This set is closed under addition, since the sum of a pair of 2 by 3 matrices is again a 2 by 3 matrix, and when such a matrix is multiplied by a real scalar, the resulting matrix is in the set also. Since M2x3( R), with the usual algebraic operations, is closed under addition and scalar multiplication, it is a real Euclidean vector space. The objects in the space—the “vectors”—are now matrices.
Since
M2x3(
R) is a vector space, what is its dimension? First, note that any 2 by 3 matrix is a unique linear combination of the following six matrices:
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Therefore, they span M2x3( R). Furthermore, these “vectors” are linearly independent: none of these matrices is a linear combination of the others. (Alternatively, the only way k1 E1 + k2 E2 + k3 E3 + k4 E4 + k5 E5 + k6 E6 will give the 2 by 3 zero matrix is if each scalar coefficient, ki , in this combination is zero.) These six “vectors” therefore form a basis for M2x3( R), so dim M2x3( R) = 6.
If the entries in a given 2 by 3 matrix are written out in a single row (or column), the result is a vector in
R6. For example,
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The rule here is simple: Given a 2 by 3 matrix, form a 6-vector by writing the entries in the first row of the matrix followed by the entries in the second row. Then, to every matrix in
M2x3(
R) there corresponds a unique vector in
R6, and vice versa. This one-to-one correspondence between
M2x3(
R) and
R6,
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is compatible with the vector space operations of addition and scalar multiplication. This means that
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The conclusion is that the spaces M2x3( R) and R6 are structurally identical, that is, isomorphic, a fact which is denoted M2x3( R) ≅ R6. One consequence of this structural identity is that under the mapping φ—the isomorphism—each basis “vector” Ei given above for M2x3( R) corresponds to the standard basis vector e i for R6. The only real difference between the spaces R6 and M2x3( R) is in the notation: The six entries denoting an element in R6 are written as a single row (or column), while the six entries denoting an element in M2x3( R) are written in two rows of three entries each.
This example can be generalized further. If m and n are any positive integers, then the set of real m by n matrices, Mmxn ( R), is isomorphic to R mn , which implies that dim Mmxn ( R) = mn.
Example 1: Consider the subset S3x3( R) ⊂ M3x3( R) consisting of the symmetric matrices, that is, those which equal their transpose. Show that S3x3( R) is actually a subspace of M3x3( R) and then determine the dimension and a basis for this subspace. What is the dimension of the subspace Snxn ( R) of symmetric n by n matrices?
Since M3x3( R) is a Euclidean vector space (isomorphic to R9), all that is required to establish that S3x3( R) is a subspace is to show that it is closed under addition and scalar multiplication. If A = AT and B = BT, then ( A + B)T = AT + BT = A + B, so A + B is symmetric; thus, S3x3( R) is closed under addition. Furthermore, if A is symmetric, then ( kA)T = kAT = kA, so kA is symmetric, showing that S3x3( R) is also closed under scalar multiplication.
As for the dimension of this subspace, note that the 3 entries on the diagonal (1, 2, and 3 in the diagram below), and the 2 + 1 entries above the diagonal (4, 5, and 6) can be chosen arbitrarily, but the other 1 + 2 entries below the diagonal are then completely determined by the symmetry of the matrix:
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Therefore, there are only 3 + 2 + 1 = 6 degrees of freedom in the selection of the nine entries in a 3 by 3 symmetric matrix. The conclusion, then, is that dim
S3x3(
R) = 6. A basis for
S3x3(
R) consists of the six 3 by 3 matrices
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In general, there are n + ( n − 1) + … + 2 + 1 = ½ n( n + 1) degrees of freedom in the selection of entries in an n by n symmetric matrix, so dim Snxn ( R) = 1/2 n( n + 1).
Polynomial spaces. A polynomial of degree
n is an expression of the form
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where the coefficients ai are real numbers. The set of all such polynomials of degree ≤ n is denoted Pn . With the usual algebraic operations, Pn is a vector space, because it is closed under addition (the sum of any two polynomials of degree ≤ n is again a polynomial of degree ≤ n) and scalar multiplication (a scalar times a polynomial of degree ≤ n is still a polynomial of degree ≤ n). The “vectors” are now polynomials.
There is a simple isomorphism between
Pn
and
R
n+1
:
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This mapping is clearly a one-to-one correspondence and compatible with the vector space operations. Therefore,
Pn
≅
R
n+1
, which immediately implies dim
Pn
=
n + 1. The standard basis for
Pn
, { 1,
x,
x2,…,
xn
}, comes from the standard basis for
R
n+1
, {
e1,
e2,
e3,…,
e
n+1
}, under the mapping φ−1:
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Example 2: Are the polynomials P1 = 2 − x, P2 = 1 + x + x2, and P3 = 3 x − 2 x2 from P2 linearly independent?
One way to answer this question is to recast it in terms of
R3, since
P2 is isomorphic to
R3. Under the isomorphism given above,
p1 corresponds to the vector
v1 = (2, −1, 0),
p2 corresponds to
v2 = (1, 1, 1), and
p3 corresponds to
v3 = (0, 3, −2). Therefore, asking whether the polynomials
p1,
p2, and
p3 are independent in the space
P2 is exactly the same as asking whether the vectors
v1,
v2, and
v3 are independent in the space
R3. Put yet another way, does the matrix
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have full rank (that is, rank 3)? A few elementary row operations reduce this matrix to an echelon form with three nonzero rows:
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Thus, the vectors—either v1, v2, v3, are indeed independent.
Function spaces. Let A be a subset of the real line and consider the collection of all real-valued functions f defined on A. This collection of functions is denoted R A . It is certainly closed under addition (the sum of two such functions is again such a function) and scalar multiplication (a real scalar multiple of a function in this set is also a function in this set), so R A is a vector space; the “vectors” are now functions. Unlike each of the matrix and polynomial spaces described above, this vector space has no finite basis (for example, R A contains Pn for every n); R A is infinite-dimensional. The real-valued functions which are continuous on A, or those which are bounded on A, are subspaces of R A which are also infinite-dimensional.
Example 3: Are the functions f1 = sin2 x, f2 = cos2 x, and f3 f3 ≡ 3 linearly independent in the space of continuous functions defined everywhere on the real line?
Does there exist a nontrivial linear combination of f1, f2, and f3 that gives the zero function? Yes: 3 f1 + 3 f2 − f3 ≡ 0. This establishes that these three functions are not independent.
Example 4: Let C2( R) denote the vector space of all realvalued functions defined everywhere on the real line that possess a continuous second derivative. Show that the set of solutions of the differential equation y” + y = 0 is a 2-dimensional subspace of C2( R).
From the theory of homogeneous differential equations with constant coefficients, it is known that the equation
y” +
y = 0 is satisfied by
y1 = cos
x and
y2 = sin
x and, more generally, by any linear combination,
y =
c1 cos
x +
c2 sin
x, of these functions. Since
y1 = cos
x and
y2 = sin
x are linearly independent (neither is a constant multiple of the other) and they span the space
S of solutions, a basis for
S is {cos
x, sin
x}, which contains two elements. Thus,
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as desired.
