Understanding Linearly Independent Vectors and Matrix Operations
Learn about linearly independent vectors, determinants, singular matrices, minors, cofactors, principal minors, inverse matrices, and the rank of a matrix. Understand the concepts and properties related to these fundamental topics in linear algebra.
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LECTURE THREE Linear independent vectors
The vectors x1, x2, . . . , xnare linearly dependent if there are scalars k1, k2, . . ., kn not all zero such that : k1x1+k2x2+ . . . +knxn= 0 Otherwise, the set of vectors are linearly independent. i.e. where it is impossible to find non-zero k1, k2, . . ., kn such that : The Determinant of a Square Matrix Associated with every square matrix there is a unique scalar number called its "determinant". The formal definition of the determinant of n n matrix A is the sum of all products consisting of one element from each row and column and multiplied by (-1) if the number of inversions of the particular permutation j1, j2, j3, , jn from the standard order 1, 2, 3, , n is odd.
Singular Matrix A square matrix is called: "singular" if its determinant is zero, and called: "non- singular" if its determinant is non-zero. Minors A minor of element (aij) of A is the determinant of the matrix formed by deleting the ithrow and the jthcolumn of A. Cofactor Aijis the minor multiplied by (-1)th. So, we can compute minor with (i=1,2,3, ,n). respect to the rows
Principal minors A minor of a matrix An nis said to be a principal minor if it is obtained by deleting certain rows and the same columns of A. Thus, the diagonal elements of a principal minor of A are the diagonal elements of A. Inverse Matrix The inverse of a square matrix A is that unique matrix A with elements such that: AA-1=A-1A = I It is possible that A-1does not exist, just as it is not to perform scalar division by zero, then A is called to be "singular".
Rank of a Matrix The rank of matrix whether square or not is defined as: the order of the largest non-zero determinant that can be calculated from the matrix. Furthermore, the rank of a matrix can be defined also as: the maximum number of linearly independent vectors (either rows or columns) in the matrix. Note: If the number of rows/ (columns) in a matrix exceeded the number of columns/ (rows), then the rank of a matrix would be the number of linearly independent columns/ (rows). i.e. r(A)(m n) min {m , n}
