Science:Math Exam Resources/Courses/MATH221/April 2013/Question 02 (b)
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Question 02 (b) 

Let
The matrices A and B are related by , where M is an invertible 4x4 matrix. b) Find the dimension and a basis of the nullspace of A. 
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Hint 

Again, is the nullspace preserved under left multiplication by invertible matrices? Can B be furthered simplified? 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. By the RankNullity theorem, the dimension of the nullspace of A is equal to the number of column minus the rank of A (which we found in part (a)), thus it is 5  3 = 2. The nullspace is unaffected by left multiplication by an invertible matrix, that is if and only if . Thus, we can use the matrix B to compute the nullspace. To find a basis of the nullspace, we will further simplify B by L_{1} → L_{1}2L_{2}, followed by L_{2} → L_{2}5L_{3} and L_{1} → L_{1}+7L_{3} to get the matrix Again, further row operations won't change the nullspace. Applying this matrix to a vector and solving for zero we get the relations We know that the dimension of the nullspace is 2, thus there must be 2 free variables, namely . So, we can read off the basis as . 