A vector space is defined (Definition VS) as a set of objects (“vectors”) endowed with a definition of vector addition ($+$) and a definition of scalar multiplication (written with juxtaposition). Many of our definitions about vector spaces involve linear combinations (Definition LC), such as the span of a set (Definition SS) and linear independence (Definition LI). Other definitions are built up from these ideas, such as bases (Definition B) and dimension (Definition D). The defining properties of a linear transformation require that a function “respect” the operations of the two vector spaces that are the domain and the codomain (Definition LT). Finally, an invertible linear transformation is one that can be “undone” — it has a companion that reverses its effect. In this subsection we are going to begin to roll all these ideas into one.
A vector space has “structure” derived from definitions of the two operations and the requirement that these operations interact in ways that satisfy the ten properties of Definition VS. When two different vector spaces have an invertible linear transformation defined between them, then we can translate questions about linear combinations (spans, linear independence, bases, dimension) from the first vector space to the second. The answers obtained in the second vector space can then be translated back, via the inverse linear transformation, and interpreted in the setting of the first vector space. We say that these invertible linear transformations “preserve structure.” And we say that the two vector spaces are “structurally the same.” The precise term is “isomorphic,” from Greek meaning “of the same form.” Let us begin to try to understand this important concept.
Two vector spaces $U$ and $V$ are isomorphic if there exists an invertible linear transformation $T$ with domain $U$ and codomain $V$, $\ltdefn{T}{U}{V}$. In this case, we write $U\isomorphic V$, and the linear transformation $T$ is known as an isomorphism between $U$ and $V$.
A few comments on this definition. First, be careful with your language (Proof Technique L). Two vector spaces are isomorphic, or not. It is a yes/no situation and the term only applies to a pair of vector spaces. Any invertible linear transformation can be called an isomorphism, it is a term that applies to functions. Second, given a pair of vector spaces there might be several different isomorphisms between the two vector spaces. But it only takes the existence of one to call the pair isomorphic. Third, $U$ isomorphic to $V$, or $V$ isomorphic to $U$? It does not matter, since the inverse linear transformation will provide the needed isomorphism in the “opposite” direction. Being “isomorphic to” is an equivalence relation on the set of all vector spaces (see Theorem SER for a reminder about equivalence relations).
In Example IVSAV we avoided a computation in $P_3$ by a conversion of the computation to a new vector space, $M_{22}$, via an invertible linear transformation (also known as an isomorphism). Here is a diagram meant to illustrate the more general situation of two vector spaces, $U$ and $V$, and an invertible linear transformation, $T$. The diagram is simply about a sum of two vectors from $U$, rather than a more involved linear combination. It should remind you of Diagram DLTA.
To understand this diagram, begin in the upper-left corner, and by going straight down we can compute the sum of the two vectors using the addition for the vector space $U$. The more circuitous alternative, in the spirit of Example IVSAV, is to begin in the upper-left corner and then proceed clockwise around the other three sides of the rectangle. Notice that the vector addition is accomplished using the addition in the vector space $V$. Then, because $T$ is a linear transformation, we can say that the result of $\lt{T}{\vect{u}_1}+\lt{T}{\vect{u}_2}$ is equal to $\lt{T}{\vect{u}_1+\vect{u}_2}$. Then the key feature is to recognize that applying $\ltinverse{T}$ obviously converts the second version of this result into the sum in the lower-left corner. So there are two routes to the sum $\vect{u}_1+\vect{u}_2$, each employing an addition from a different vector space, but one is “direct” and the other is “roundabout”. You might try designing a similar diagram for the case of scalar multiplication (see Diagram DLTM) or for a full linear combination.Checking the dimensions of two vector spaces can be a quick way to establish that they are not isomorphic. Here is the theorem.
Suppose $U$ and $V$ are isomorphic vector spaces. Then $\dimension{U}=\dimension{V}$.
The contrapositive of Theorem IVSED says that if $U$ and $V$ have different dimensions, then they are not isomorphic. Dimension is the simplest “structural” characteristic that will allow you to distinguish non-isomorphic vector spaces. For example $P_6$ is not isomorphic to $M_{34}$ since their dimensions (7 and 12, respectively) are not equal. With tools developed in Section VR we will be able to establish that the converse of Theorem IVSED is true. Think about that one for a moment.