Module 4 : Solving Linear Algebraic Equations
Section 12: Excercise
 
12 Decide whether vector b belongs to column space spanned by MATH

  1. MATH

  2. MATH any $\QTR{bf}{b}$
13 Find dimension and construct a basis for the four fundamental subspaces associated with each of the matrices. MATHMATH
14 Find a non -zero vector MATH orthogonal to all rows of MATH(In other words, find Null space of matrix $\QTR{bf}{A}.)$ If such a vector exits, can you claim that the matrix is singular? Using above A matrix find one possible solution $\QTR{bf}{x}$ for MATH when MATH. Show that if vector x is a solution of the system MATH, then (MATH is also a solution for any scalar $\alpha $, i.e. MATHAlso, find dimensions of row space and column space of A.
15 If product of two matrices yields null matrix, i.e. $\QTR{bf}{A}B=[0]$, show that column space of B is contained in null space of A and the row space of A is in the left null space of B.