Linear Algebra

Linear Transformation Consider a 3 dimension system. Linear transformation in 3D along with its matrix form is given by Note: If Y=AX and Z=BY,then Z=(BA) X=Composite Linear Transformation. Topic Linear Transformation Linear Transformation Problems Show that the transformation y_1=x_1+2x_2+5x_3 ; y_2=2x_1+4x_2+11x_3 ; y_3= -2x_2+2x_3 is regular. Write down the inverse transformation Show that the transformation y_1=2x-2y-z ; y_2= - 4x+5y+3z ; y_3=x-y-z is regular and find out the Inverse transformation. Show that the transformation y_1=2x+y+z ; y_2=x+y+2z; y_3=x-2z is regular.…

Reducing Quadratic form to Canonical form by Orthogonal transformation Topic Orthogonal Transformation Problems Reduce 8x^2+7y^2+3z^2-12xy+4xz-8yz into canonical form using orthogonal transformation. Also indicate the nature, index, rank and signature of the quadratic form Reduce the Quadratic form 3x^2+5y^2+3z^2-2yz+2zx-2xy to the Canonical form  

Rayleigh's Power Method Rayleigh's power method is an iterative method to determine the numerically largest eigen value (dominant eigen value) and the corresponding eigen vector of a square matrix. Steps for solving problems Topic Rayleigh's Power Method Rayleigh's Power Method Problems Find the largest Eigen value and the corresponding Eigen vector of the matrix by the power method given that A =[(2,0,1),(0,2,0),(1,0,2)] Find the dominant Eigen value and the corresponding Eigen vector of the matrix A=[(6,-2,2),((-2,3,-1),(2,-1,3)] by power method taking…

Diagonalisation of a Square Matrix   Topic Diagonalisation of a Square Matrix Related Problems Reduce the matrix A=[(-1,3),(-2,4)] to the diagonal form Diagonalize the matrix [(-19,7),(-42,16)] Reduce the matrix A=[(5,4),(1,2)] into the diagonal form Diagonalize the matrix [(1,1),(3,-1)]  

Gauss elimination method problems Solve by Gauss elimination method 2x+y+4z=12; 4x+11y-z=33; 8x-3y+2z=20 Solve the following equations by Gauss elimination method x+2y+z=3; 2x+3y+2z=5; 3x-5y+5z=2 Solve the following system of equations by Gauss elimination method x_1+x_2+x_3+x_4=2; 2x_1-x_2+2x_3-x_4=-5; 3x_1+2x_2+3x_3+4x_4=7; x_1-2x_2-3x-3+2x_4=5; Solve by Gauss Elimination method x_1-2x_2+3x_3=2 ; 3x_1-x_2+4x_3=4 ; 2x_1+x_2-2x_3=5 Solve by Gauss-Elimination method x+y+z=6; x+2y+3z=10; x+2y+2z=8   Gauss Jordan method problems Solve the following system of equations by Gauss Jordan method x+y+z=8 ; -x-y+2z=-4 ; 3x+5y-7z=14 Solve by Gauss Jordan method the…

Row Operations explained with an example   Topics Rank of a Matrix Row Operations Related Problems Find the rank of the matrix A=[(2,1,3,5),(4,2,1,3),(8,4,7,13),(8,4,-3,-1)] by reducing it to echelon form Find the rank of the matrix [(1,1,1,6),(1,-1,2,5),(3,1,1,8),(2,-2,3,7)] Find the rank of the matrix A=[(0,1,-3,-1),(1,0,1,1),(3,1,0,2),(1,1,-2,0)] by reducing it to echelon form Find the rank of the matrix [(4,0,2,1),(2,1,3,4),(2,3,4,7),(2,3,1,4)] Find the rank of the matrix [(2,-1,-3,-1),(1,2,3,-1),(1,0,1,1),(0,1,1,-1)]  

Rank of a Matrix Elementary operations associated with a matrix Following are the elementary row transformations (can be applied for columns also) If matrix A gets transferred into another matrix B by applying any of the above transformation, then A is said to be equivalent to B,(A~B) Echelon form A matrix A of order m×n is said to be in a row reduced echelon form if 1. The leading element (the first non-zero entry )of each row is unity .…