It is proved that because k is a full rank square matrix, it is reversible, and there is an inverse of k, and both sides of the equation are multiplied by the inverse of k at the same time, so.
K inverse () = (), the first bracket is the β vector group, and the second bracket is the α vector group.
This shows that the alpha vector group can be expressed linearly by the beta vector group, so the two vector groups can be expressed linearly with each other, so the two vector groups are equivalent. Because the ranks of equivalent vector groups are the same, so are the ranks of β vector groups, so β vector groups are linearly independent.
Extended data:
In linear algebra, the column rank of matrix A is the maximum number of linearly independent columns of A. Similarly, the row rank is the maximum number of linearly independent rows of A, that is, if the matrix is regarded as a row vector or a column vector, the rank is the rank of these row vectors or column vectors, that is, the number of vectors contained in the largest independent group.
Theorem: The row rank, column rank and rank of a matrix are all equal.
Theorem: Elementary transformation does not change the rank of matrix.
Theorem: If A is reversible, then r(AB)=r(B), r(BA)=r(B).
Theorem: Rank Rab
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