MATH221 April 2013
• Q1 • Q2 (a) • Q2 (b) • Q2 (c) • Q2 (d) • Q3 • Q4 • Q5 • Q6 • Q7 (a) • Q7 (b) • Q7 (c) • Q8 • Q9 • Q10 • Q11 • Q12 (a) • Q12 (b) • Q12 (c) •
Question 02 (b)
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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Again, is the nullspace preserved under left multiplication by invertible matrices?
Can B be furthered simplified?
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By the Rank-Nullity 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 L1 → L1-2L2, followed by L2 → L2-5L3 and L1 → L1+7L3 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 .
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