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 *M* _{2x3}( **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 *M* _{2x3}( **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 *M* _{2x3}( **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:

Therefore, they span *M* _{2x3}( **R**). Furthermore, these “vectors” are linearly independent: none of these matrices is a linear combination of the others. (Alternatively, the only way *k* _{1} *E* _{1} + *k* _{2} *E* _{2} + *k* _{3} *E* _{3} + *k* _{4} *E* _{4} + *k* _{5} *E* _{5} + *k* _{6} *E* _{6} will give the 2 by 3 zero matrix is if each scalar coefficient, *k _{i} *, in this combination is zero.) These six “vectors” therefore form a basis for

*M*

_{2x3}(

**R**), so dim

*M*

_{2x3}(

**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 **R** ^{6}. For example,

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 *M* _{2x3}( **R**) there corresponds a unique vector in **R** ^{6}, and vice versa. This one‐to‐one correspondence between *M* _{2x3}( **R**) and **R** ^{6},

is compatible with the vector space operations of addition and scalar multiplication. This means that

The conclusion is that the spaces *M* _{2x3}( **R**) and **R** ^{6} are *structurally identical*, that is, **isomorphic**, a fact which is denoted *M* _{2x3}( **R**) ≅ **R** ^{6}. One consequence of this structural identity is that under the mapping ϕ—the *isomorphism*—each basis “vector” *E _{i} *given above for

*M*

_{2x3}(

**R**) corresponds to the standard basis vector

**e**

_{i }for

**R**

^{6}. The only real difference between the spaces

**R**

^{6}and

*M*

_{2x3}(

**R**) is in the notation: The six entries denoting an element in

**R**

^{6}are written as a single row (or column), while the six entries denoting an element in

*M*

_{2x3}(

**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, *M _{mxn} *(

**R**), is isomorphic to

**R**

^{mn }, which implies that dim

*M*(

_{mxn}**R**) =

*mn*.

**Example 1**: Consider the subset *S* _{3x3}( **R**) ⊂ *M* _{3x3}( **R**) consisting of the symmetric matrices, that is, those which equal their transpose. Show that *S* _{3x3}( **R**) is actually a subspace of *M* _{3x3}( **R**) and then determine the dimension and a basis for this subspace. What is the dimension of the subspace *S _{nxn} *(

**R**) of symmetric

*n*by

*n*matrices?

Since *M* _{3x3}( **R**) is a Euclidean vector space (isomorphic to **R** ^{9}), all that is required to establish that *S* _{3x3}( **R**) is a subspace is to show that it is closed under addition and scalar multiplication. If *A* = *A* ^{T} and *B* = *B* ^{T}, then ( *A + B*) ^{T} = *A* ^{T} + *B* ^{T} = *A + B*, so *A + B* is symmetric; thus, *S* _{3x3}( **R**) is closed under addition. Furthermore, if *A* is symmetric, then ( *kA*) ^{T} = *kA* ^{T} = *kA*, so *kA* is symmetric, showing that *S* _{3x3}( **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:

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 *S* _{3x3}( **R**) = 6. A basis for *S* _{3x3}( **R**) consists of the six 3 by 3 matrices

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 *S _{nxn} *(

**R**) = 1/2

*n*(

*n*+ 1).

**Polynomial spaces**. A polynomial of degree *n* is an expression of the form

where the coefficients *a _{i} *are real numbers. The set of all such polynomials of degree ≤

**is denoted**

*n**P*. With the usual algebraic operations,

_{n}*P*is a vector space, because it is closed under addition (the sum of any two polynomials of degree ≤

_{n}*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 *P _{n} *and

**R**

^{n+1 }:

This mapping is clearly a one‐to‐one correspondence and compatible with the vector space operations. Therefore, *P _{n} *≅

**R**

^{n+1 }, which immediately implies dim

*P*=

_{n}*n*+ 1. The standard basis for

*P*, { 1,

_{n}*x*,

*x*

^{2},…,

*x*}, comes from the standard basis for

^{n}**R**

^{n+1 }, {

**e**

_{1},

**e**

_{2},

**e**

_{3},…,

**e**

_{n+1 }}, under the mapping ϕ

^{−1}:

**Example 2**: Are the polynomials **P** _{1} = 2 − *x*, **P** _{2} = 1 + *x* + *x* ^{2}, and **P** _{3} = 3 *x* − 2 *x* ^{2} from *P* _{2} linearly independent?

One way to answer this question is to recast it in terms of **R** ^{3}, since *P* _{2} is isomorphic to **R** ^{3}. Under the isomorphism given above, **p** _{1} corresponds to the vector **v** _{1} = (2, −1, 0), **p** _{2} corresponds to **v** _{2} = (1, 1, 1), and **p** _{3} corresponds to **v** _{3} = (0, 3, −2). Therefore, asking whether the polynomials **p** _{1}, **p** _{2}, and **p** _{3} are independent in the space *P* _{2} is exactly the same as asking whether the vectors **v** _{1}, **v** _{2}, and **v** _{3} are independent in the space **R** ^{3}. Put yet another way, does the matrix

have full rank (that is, rank 3)? A few elementary row operations reduce this matrix to an echelon form with three nonzero rows:

Thus, the vectors—either **v** _{1}, **v** _{2}, **v** _{3}, 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 *P _{n} *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 **f** _{1} = sin ^{2} *x*, **f** _{2} = cos ^{2} *x*, and **f** _{3} **f** _{3} ≡ 3 linearly independent in the space of continuous functions defined everywhere on the real line?

Does there exist a nontrivial linear combination of **f** _{1}, **f** _{2}, and **f** _{3} that gives the zero function? Yes: 3 **f** _{1} + 3 **f** _{2} − **f** _{3} ≡ **0**. This establishes that these three functions are not independent.

**Example 4**: Let *C* ^{2}( **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 *C* ^{2}( **R**).

From the theory of homogeneous differential equations with constant coefficients, it is known that the equation *y*” + *y* = 0 is satisfied by *y* _{1} = cos *x* and *y* _{2} = sin *x* and, more generally, by any linear combination, *y* = *c* _{1} cos *x* + *c* _{2} sin *x*, of these functions. Since *y* _{1} = cos *x* and *y* _{2} = 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,

as desired.