Week 1 Summary

COMP116 Lectures Summaries 2021-02-05

Number

1. Recall the different classes of number and recognise which is the most suitable in different contexts.
2. Be aware of how the call of real number originates and the idea of the subset of reals obtains as roots of polynomial expressions.

Typical Questions

You may be asked which type of number is the most suitable for a given application.

Polynomial Forms

You should be aware of:

1. The formal definition of polynomial of degree $k$ in $x$.
2. The notions of coefficient and the set $H_k[X]$ for numbers $H$.
3. Basic operations including addition and scalar multiplication.
4. The concept of roots of a polynomial properties.

You don’t need to know:

1. Root finding algorithms or methods.
2. How to multiply, factor or divide polynomials.

Typical Questions

1. Asked for the coefficient after expanding.
2. Convert a wordy question into a polynomial.
3. Ask for the degree of an expanded polynomial.

Vectors and Matrices

You should be aware of:

1. The form and attributes of $n$-vectors: direction, size. Component properties (ordered number types used).
2. Basic vector operations: addition, scalar multiple, size.
3. The properties and requirements of vector spaces.
4. The form take by matrices and simpler operations/.
5. The process of multiply a matrix by a vector.
6. The notions of vector space dimension , linear combination, linear independence and basis of vector space.

You don’t need to know:

1. The formal structure of vector product definitions: scalar, dot.
2. The matric concept of determinant, singularity, inverse.
3. How to calculate the product of matrices larger than 4.

Typical Questions.

1. What is the size of a vector.

2. The result of transformations.

3. Whether vectors are lineally independent.

   Weak point.

4. Show that a vector can be described as a linear combination of the others if it is not independent.

   Weak point.

Linear Transformation and Graphics

You should be aware of:

1. How linear transformations and matrix-vector are related.

2. What’s needed for a mapping to be linear transformation.

   Weak point.

3. The structure of $2\times2$ and $3\times3$ scaling matrices.

4. The concept of homogeneous coordinates and matrices.

5. The way in which matrix produces are combines to realise different graphics effects.

6. The consequences of effect not being commutative.

Typical Questions

1. What the effect of applying a particular combination of matrices will be to a point in space.
2. How are a given sequence of operations realised by a sequence of matrix-vector products.
3. You may need to expand on the previous with additional comments.