Matrix Algebra - Dimension and Rank
5 important questions on Matrix Algebra - Dimension and Rank
The dimension of a nonzero subspace H, denoted by dim H, is?
The dimension of the zero subspace {0}
The rank of a matrix A, denoted by rank A, is?
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Theorem 15 The Basis Theorem
Let H be a p-dimensional subspace of Rn. Any linearly independent set of exactly p elements in H is?
Theorem The Invertible Matrix Theorem (continued)
Let A be an n x n matrix. Then the following statements are each equivalent tothe statement that A is an invertible matrix.
m. The columns of A form a basis of Rn.
n. Col A = Rn
o. dim Col A = n
p. rank A = n
q. Nul A = {0}
r. dim Nul A = 0
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