Science:Math Exam Resources/Courses/MATH152/April 2016/Question A 27/Solution 1

The null space of A is spanned by $[0,1,0,1]^{T},[7,0,1,0]^{T}$

By Rank–nullity theorem, the rank of A is 2. In the other words, the dimension of space spanned by column vectors are 2.

The row reduced echelon form of a matrix is unique. There are two pivot columns. The range of A is the span of column space. From the kernels, we know the third

column are seven multiples of the first column, and the fourth column are -1 multiple of the second column of A.

Thus, the row reduced echelon form is ${\begin{bmatrix}1&0&-7&0\\0&1&0&-1\\0&0&0&0\\\end{bmatrix}}$