Dimension Theorem

• Let and be vector spaces, and be linear. The null space or kernel is the set

  The column space of is defined as

• (Friedberg 2.1) Let and be vector spaces and be linear. Then and are subspaces of and respectively.

• (Friedberg 2.2) Let and be vector spaces and be linear. If has a basis then

• If and are finite dimensional, then we define the Nullity of and the Rank of as

• (Friedberg 2.3) Dimension Theorem. Let and be vector spaces and let be linear. If is finite dimensional then

  • Extending the basis of yields elements which, when applying the linear transformation, are in the span of . In fact, they form a basis of

• (Friedberg 2.4) is one to one if and only if

• (Friedberg 2.5) Let and be vector spaces of equal finite dimension and be linear. Then is one-to-one if and only if is onto.

• (Friedberg 2.6) Suppose is finite-dimensional with basis . For any vectors , there exists exactly one linear transformation such that for

  In other words: The effect of a linear transformation is completely determined by its effect on the basis of the domain.

  • (Friedberg 2.6.1) Let be vector spaces and suppose has a finite basis . If are linear, and for then .
  • (Friedberg e2.1.27) Let be possibly infinite-dimensional spaces over a common field and be a basis for . For Any function , there exists exactly one linear transformation such that

• (Friedberg e2.4.18, Friedberg 3.3) Let be a linear transformation from an -dimensional vector space to an -dimensional vector space . Let be ordered bases for respectively. Let . Then

  • Intuition: This follows because linear transformations are isomorphic to matrices. Hence, vector subspaces are preserved and so are the Null Space and Column Space.

    Hence, we can define ranks of matrices in an analogous manner to our definition for linear transformations

• Let be a block matrix of the form

  Then

  A special case is if in which case

Links §

• Friedberg, Insel and Spence