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5. Let A represent an m × n matrix and B an m × p matrix. Then, (1) C(A) ⊂ C(B) if and only if R(A ) ⊂ R(B ), and (2) C(A) C(B) if and only if R(A ) R(B ). 3 Bases a. Some definitions (and notation) The span of a finite set of matrices (having the same dimensions) is defined as follows: the span of a finite nonempty set {A1 , . . , Ak } is the set consisting of all matrices that are expressible as linear combinations of A1 , . . , Ak , and the span of the empty set is the set {0}, whose only element is the null matrix.

Ak } of two or more m × n matrices is linearly dependent if and only if at least one of the matrices is expressible as a linear combination of the others; that is, if and only if, for some integer j (1 ≤ j ≤ k), Aj is expressible as a linear combination of A1 , . . , Aj −1 , Aj +1 , . . , Ak . Proof. Suppose that, for some j , Aj is expressible as a linear combination Aj x1 A1 + · · · + xj −1 Aj −1 + xj +1 Aj +1 + · · · + xk Ak of the other k − 1 m × n matrices. Then, (−x1 )A1 + · · · + (−xj −1 )Aj −1 + Aj +(−xj +1 )Aj +1 + · · · + (−xk )Ak 0, implying that {A1 , A2 , .

Any two bases for the same linear space contain the same number of matrices. The number of matrices in a basis for a linear space V is called the dimension of V and is denoted by the symbol dim V or dim(V). 2, we obtain the results expressed in the following three theorems. 7. If a linear space V is spanned by a set of r matrices, then dim V ≤ r, and if there is a set of k linearly independent matrices in V, then dim V ≥ k. 8. If U is a subspace of a linear space V, then dim U ≤ dim V. 9. Any set of r linearly independent matrices in an r-dimensional linear space V is a basis for V.