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.

Exercise 2 - Amdahl's Law

A programmer has parallelized 99% of a program, but there is no value in increasing the problem size, i.e., the program will always be run with the same problem size regardless of the number of processors or cores used. What is the expected speedup on 20 processors? Solution As per Amdahl's law, the speedup, N - No of processors = 20 f - % of parallel operation = 99% = 1 / (1 - 0.99) + (0.99 / 20) = 1 / 0.01 + (0.99 / 20) = 16.807 The expected speedup on 20 processors is 16.807

Exercise 1 - Amdahl's Law

A programmer is given the job to write a program on a computer with processor having speedup factor 3.8 on 4 processors. He makes it 95% parallel and goes home dreaming of a big pay raise. Using Amdahl’s law, and assuming the problem size is the same as the serial version, and ignoring communication costs, what is the speedup factor that the programmer will get? Solution Speedup formula as per Amdahl's Law, N - no of processor = 4 f - % of parallel operation = 95% Speedup = 1 / (1 - 0.95) + (0.95/4) = 1 / 0.5 + (0.95/4) Speedup = 3.478 The programmer gets 3.478 as t he speedup factor.

Minor, Cofactor, Determinant, Adjoint & Inverse of a Matrix

Consider a matrix Minor of a Matrix I n the above matrix A, the minor of first element a 11 shall be Cofactor The Cofactor C ij of an element a ij shall be When the sum of row number and column number is even, then Cofactor shall be positive, and for odd, Cofactor shall be negative. The determinant of an n x n matrix can be defined as the sum of multiplication of the first row element and their respective cofactors. Example, For a 2 x 2 matrix Cofactor C 11 = m 11 = | a 22 | = a 22 = 2 Determinant The determinant of A is |A| = (3 x 2) - (1 x 1) = 5 Adjoint or Adjucate The Adjoint matrix of A , adjA is the transpose of its cofactor matrix. Inverse Matrix A matrix should be square matrix to have an inverse matrix and also its determinant should not be zero. The multiplication of matrix and its inverse shall be Identity matrix. The square matrix has no inverse is called Singular. Inv A = adjA / |A| [ adjoint A / determ...

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