Row Space, Column Space & Null Space

Consider A_mXn matrix,

The row vectors, r of A,

r₁ = (a₁₁ .... a_1n), .................... , r_m = (a_m1 ......... a_mn)

The column vectors, c of A,

The subspace span { r₁ ..... r_m } of Rⁿ, where r₁, ......., r_m are the rows of the matrix A, is called the row space of A.

The subspace span { c₁, ...... , c_n } of R^m, where c_{1, .....,}c_n are the columns of the matrix A, is called the column space of A.

The solution space of the system of homogeneous linear equations Ax = 0 is called the nullspace of A.

To find a basis for the row space of A, the row echelon form of A to be derived, and consider the non-zero rows that result from this reduction. These non-zero rows are linearly independent.

To find a basis for the column space of A, the row echelon form of A to be derived, and consider the pivot columns that result from this reduction. These pivot columns are linearly independent, and that any non-pivot column is a linear combination of the pivot columns.

The dimension of the row space of A is equal to the number of non-zero rows in the row echelon form. The dimension of the column space of A is equal to the number of pivot columns in the row echelon form. The nullity of a matrix is the dimension of its nullspace.

For any matrix A with entries in R, the dimension of the row space is the same as the dimension of the column space.

The rank of a matrix A is equal to the common value of the dimension of its row space and the dimension of its column space.

For any matrix A with entries in R, the sum of the dimension of its column space and the dimension of its nullspace is equal to the number of columns of A, known as the Rank-nullity theorem.

Matrix Eigenvalues & Eigenvectors

The process of finding an unknown scalar, λ and a nonzero vector, x for a given non zero square matrix, A of dimension n x n is called matrix eigenvalue or eigenvalue. Ax = λ x The λ and x which satisfies the above equation is called eigen value and eigenvector. Ax should be proportional to x. The multiplication will produce a new vector that will have the same or opposite direction as the original vector. The set of all the eigenvalues of A is called the spectrum of A. The largest of the absolute values of the eigenvalues of A is called the spectral radius of A. To determine eigenvalue and eigenvector, the equation can be written in matrix notation, (A - λI)x = 0 By Cramer's theorem, the homogeneous linear system of equations has a nontrivial solution if and only if the corresponding determinant of the coefficients is zero. A - λI is called characteristic matrix and D( λ) is characteristic determinant of A. The above equation is called characteristic ...

Gaussian Elimination - Row reduction Algorithm

Gaussian elimination is a method for solving matrix equations of the form, Ax=b. This method is also known as the row reduction algorithm. Back Substitution Solving the last equation for the variable and then work backward into the first equation to solve it. The fundamental idea is to add multiples of one equation to the others in order to eliminate a variable and to continue this process until only one variable is left. Pivot row The row that is used to perform elimination of a variable from other rows is called the pivot row. Example: Solving a linear equation The augmented matrix for the above equation shall be The equation shall be solved using back substitution. The eliminating the first variable (x1) in the first row (Pivot row) by carrying out the row operation. As the second row become zero, the row will be shifted to bottom by carrying out partial pivoting. Now, the second variable (x2) shall be eliminated by carrying out the row operation again. ...

Decision Tree - Gini Index

The Gini index is used in CART. The Gini index measures the impurity of the data set, where p i - probability that data in the data set, D belong to class, C i and pi = |C i,D |/|D| There are 2 v - 2 possible ways to form two partitions of the data set, D based on a binary split on a attribute. Each of the possible binary splits of the attribute is considered. The subset that gives the minimum Gini index is selected as the splitting subset for discrete valued attribute. The degree of Gini index varies between 0 and 1. The value 0 denotes that all elements belong to a certain class or if there exists only one class, and the value 1 denotes that the elements are randomly distributed across various classes. A Gini Index of 0.5 denotes equally distributed elements into some classes. The Gini index is biased toward multivalued attributes and has difficulty when the number of classes is large.

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