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MA10210: Algebra 1B

MA10210: Algebra 1B. http://people.bath.ac.uk/aik22/ma10210. Comments on Sheet 8. When finding determinants of matrices by expanding, find a row or column with zeros. Especially when working with big matrices, this can be much more efficient depending on the matrix.

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MA10210: Algebra 1B

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  1. MA10210: Algebra 1B http://people.bath.ac.uk/aik22/ma10210

  2. Comments on Sheet 8 • When finding determinants of matrices by expanding, find a row or column with zeros. • Especially when working with big matrices, this can be much more efficient depending on the matrix. • Revise matrix rank and null spaces – some solutions were using the wrong definitons.

  3. Comments on Sheet 8 • Don’t guess and waffle. • When trying to get ideas on how to tackle a question, writing down everything you can think of can be very helpful. • When trying to understand an explanation, unnecessary ideas make it harder to figure out what’s going on. Either rewrite the solution with only the necessary bits, or make it clear what the relevant facts are. • Don’t just write a list of random facts in the hope whoever’s marking will take pity. They won’t.

  4. Warm-up Question • Q1: • (i) Find determinant of , find eigenvalues. • (ii) Use this to find the eigenvectorssatisfying . • Use the eigenvectors to find P. • (iii) Having diagonalised A, find An=PDnP. • Q4: • Note: eigenvalues/eigenvectors satisfy Av=λv.

  5. Overview of Sheet 9 • Q2: similar to Q1 • Q3: find a matrix A such that vn=Anv0. • Q5: (i) what does it mean for 0 to be an eigenvalue? • Q5: (ii) use the fact that Av=λv again. • Q6: (i) consider det(B). • Q6: (ii) consider Pv.

  6. Overview of Sheet 9 • Q7: • Show that zero is an eigenvalue • Show that if is a is an eigenvalue, then a must be 0 • Find some N such that φN-1(v) ≠ 0 • What happens if you apply φ to this map?

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