Comments on Sheet 6 Dont confuse dimension and rank Theyre closely related but not the same: A vector space has dimension. A map has rank. Neither is cardinality, so dont use the notation |V| Not every map is surjective
Comments on Sheet 7 Q6 – this question is asking for an alternative proof to Prop. 4.1.2: If we allow use of 4.1.2, theres nothing left in the question. The hints suggest an alternative method. Dont divide by something that might be zero: show its not zero first In a field, ab0 is enough to show that neither a nor b is zero.
Warm-up Question Q1: Find determinant by expanding about a row Find determinant by expanding about a column (Check theyre the same) Find the inverse Matrix of minors Cof(A) Adj(A) A -1
Overview of Sheet 8 Q2: similar to Q1 Q3: A invertible if det A 0 Q4: put in matrix form, use notes on Cramers rule Q5(i): use hint; Q5(ii): note that A is invertible if and only if the columns of A are a basis.
Overview of Sheet 8 Q6(i): the determinant is zero if two (or more) rows are equal. When are two rows equal? Consider the determinant as a polynomial in x, y, z. What is the degree of this polynomial? Q6(ii): find the link to Q6(i), writing the system in matrix form Q6(iii): use some of the ideas from Q6(i)