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Dimension With the exception of the linear space {0}, for which the only basis is the empty set, every linear space of m × n matrices has an infinite number of bases. 2 that if two sets, one containing k1 matrices and the other containing k2 matrices, are both bases for the same linear space, then k1 ≤ k2 and k2 ≤ k1 . Thus, we have the following result. 6. 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).

1 1. For what values of the scalar k are the three row vectors (k, 1, 0), (1, k, 1), and (0, 1, k) linearly dependent, and for what values are they linearly independent? Describe your reasoning. 2 2. Let A, B, and C represent three linearly independent m×n matrices. Determine whether or not the three pairwise sums A + B, A + C, and B + C are linearly independent. (Hint. ) 4 Linear Spaces: Row and Column Spaces Associated with any matrix is a very important characteristic called the rank. 4. There are several (consistent) ways of defining the rank.

Ap } spans V, so does the set {A1 , . . , Ap , B1 , . . , Bq }. Moreover, if the set {A1 , . . , Ap , B1 , . . , Bq } spans V and if B1 , . . , Bq are expressible as linear combinations of A1 , . . , Ap , then the set {A1 , . . , Ap } spans V. 1 can be proved rather simply by showing that if B1 , . . , Bq are expressible as linear combinations of A1 , . . , Ap , then any linear combination of the matrices A1 , . . , Ap , B1 , . . , Bq is expressible as a linear combination of A1 , .