Linear Combination

• Let be a Vector Space and a nonempty subset of . A vector is a linear combination of elements in if there exists a finite number of elements and scalars such that

• (Fraleigh 30.16) Let be a vector space over field and . If is a linear combination of vectors and each , is a linear combination of vectors . Then is a linear combination of the .

Linear Independence §

• A subset of a vector space is said to be linearly dependent if there exists a finite number of distinct vectors and scalars such that

  Otherwise, and its elements are said to be linearly independent

• Any subset containing the zero vector must be linearly dependent

• The following are true about linear independence for any vector space

  • is linearly independent.
  • A set consisting of a single non-zero vector is linearly independent.
  • For a linearly independent set, the only way to get the zero vector is by setting

• (Friedberg 1.6) Let . If is linearly dependent, then so is

  An immediate corollary: Let . If is linearly independent, then so is

• A maximal linearly independent subset of is a subset of such that

  • is linearly independent
  • Any subset of that properly contains is linearly dependent.

Span and Basis §

• (Friedberg 1.5) If is a nonempty subset of a vector space , then the set consisting of all linear combinations of elements of is a subspace of . Moreover, is the smallest subspace of containing .

  We call the span of , denoted .

  We define

• A subset of a vector space generates if .

• A basis for a vector space is a linearly independent subset of that generates . We say that the elements of form a basis for . A basis can be ordered if the order of the elements in the basis is important.

  We can generalize this definition to say that a basis is a maximal linearly independent subset of (see Friedberg 1.12)

• (Friedberg 1.7) Let be a vector space and . Then is a basis for if and only if each vector can be uniquely expressed as a linear combination of the form

• (Friedberg 1.8) Let be a linearly independent subset of a vector space that is not in . Then if and only if

• (Friedberg 1.9; Fraleigh 30.18) If is generated by a finite set , then a subset of is a basis for . Hence, has a finite basis.

• (Friedberg 1.10) Basis Replacement Theorem / Steinitz Replacement Lemma Let be a vector space having a basis containing exactly elements. Let be a linearly independent subset of containing elements.

  There exists containing exactly elements such that

  In other words, each element in the basis is replaceable.

  • (Friedberg 1.10.1) Any linearly independent subset of containing exactly elements is a basis for .
  • (Friedberg 1.10.2) Any subset containing more than elements is linearly dependent. Any linearly independent subset of contains at most elements
  • (Friedberg 1.10.3; Fraleigh 30.20) Every basis for contains exactly elements.
  • (Friedberg 1.10.4) Let be a subset of such that and has at most elements, then is a basis for .
  • (Friedberg 1.10.5; Fraleigh 30.19) Extension Property Let be a linearly independent subset of . Then, there exists such that is a basis for . Every linearly independent subset of can be extended to a basis for

• The dimension of a vector space is the unique number of elements in each basis of , denoted . A vector space is finite-dimensional if the dimension is finite and infinite-dimensional otherwise.

• (Friedberg e1.6.19,; Fraleigh 30.17) Let , and . Then a subset of is a basis for . Moreover, contains at least at least elements.

• (Friedberg 1.12) Let be a vector space and a subset that generates . If is a maximal linearly independent subset of , then is a basis for .

• (Friedberg 1.13) Let be a linearly independent subset of a vector space . There exists a maximal linearly independent subset of that contains .

  Therefore, alongside (Friedberg 1.12) Every vector space has a basis.

Links §

• Linear Algebra by Friedberg Insel and Spence
• Fraleigh