### Basis

##### 3. Isomorphism

The coordinates with regard to a basis B = {b1, b2, ..., bn} of a vector space V may be considered as a transformation [?]B: VRn. It is easy to verify that [?]B is an invertible linear transformation with

T(x1, x2, ..., xn) = x1b1 + x2b2 + ... + xnbn: RnV

as the inverse.

Two vector spaces are isomorphic if there is an invertible linear transformation between them. Any such invertible linear transformation is an isomorphism.

In particular, if a vector space has a finite basis, then it is isomorphic to the euclidean space. We also note that if a vector space is spanned by finitely many vectors, then it has a finite basis. Such vector spaces are called finite dimensional.

Example The coordinates with regard to the basis {1, t, t2, ..., tn} is the isomorphism

a0 + a1t + a2t2 + ... + antn ↔ (a0, a1, a2, ..., an).

This has been used in converting some problems about polynomials to euclidean vectors.

The coordinates with regard to the basis

 { [ 1 0 ], [ 0 1 ], [ 0 0 ], [ 0 0 ] } 0 0 0 0 1 0 0 1

is the isomorphism

 [ a b ] ↔ (a, b, c, d). c d

This has been used in converting some problems about matrices to euclidean vectors.

Example By an earlier example, the coordinates with regard to the basis {(3, 1, 2), (1, -1, 2), (-1, 1, 1)} gives us the isomorphism

(b1, b2, b3) ↔ ((1/4)b1 + (1/4)b2, - (1/12)b1 - (5/12)b2 + (1/3)b3, - (1/3)b1 + (1/3)b2 + (1/3)b3).

This is the inverse of the isomorphism

(x1, x2, x3) ↔ (3x1 + x2 - x3, x1 - x2 + x3, 2x1 + 2x2 + x3),

whose matrix has the given basis vectors as columns.

Given an isomorphism, linear algebra problems (definitions, concepts, theories, proofs, etc) in one vector space is equivalent to linear algebra problems (definitions, concepts, theories, proofs, etc) in another vector space. In particular, for finite dimensional vector spaces, the coordinates identify the linear algebra in general vector spaces with the linear algebra in the euclidean space. Consequently, we have

Any linear algebraic fact about euclidean spaces also holds in general (finite dimensional) vector spaces.

For example, the following is the translation of this fact about euclidean spaces into general vector spaces. (However, see this exercise and the remark)

For a linear transformation T: VW, the following are equivalent.

• T is invertible.
• T is onto and one-to-one.
• rangeT = W, kernelT = {0}.

The concept of isomorphism can be extended to any equivalence between two systems. For example, different languages are largely isomorphic, with the isomorphisms given by dictionaries. Suppose a French asks a Spanish to do some work, the Spanish may translate the job description into Spanish, finish the job in a Spanish environment, and then report back in French after finishing the job. For another example, the numbers people use may vary in notation, the base (binary, decimal), etc. However, they are all equivalent number systems. The same also applies to the measurement systems such as length (meter, inch), weight (gram, pound), money (US dollar, Hong Kong dollar, Yen, Yuan, Euro).