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Recap of linear algebra: vectors, matrics , transformations, …

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Recap of linear algebra: vectors, matrics, transformations, …

Background knowledge for 3DM

Marc van Kreveld

- A vector is an ordered pair, triple, … of (real) numbers, often written as a column
- A vector (3, 4) can be interpreted as the point with x-coordinate 3 and y-coordinate 4, so (3, 4) as well
- A vector like (2, 1, –4) can be interpreted as a point in 3-dimensional space

Three times the vector (3, 2), and the point (3, 2)

- The length of a vector (a, b) is denoted |(a, b)| and is obtained by the Pythagoras Theorem:
- The length of a vector (a, b, c) is denoted |(a, b, c)| and is given by : Be aware of length and dimensionality and their difference

- Two vectors of the same dimensionality can be added; just add the corresponding components: (a,b) + (c,d) = (a+c, b+d)
- The result is a vector of the same dimensionality
- Geometric interpretation: place one arrow’s start at the end of the other, and take the resulting arrow

purple + purple = blue

- A value is also called a scalar
- We can multiply a scalar k with a vector (a, b); this is defined to be the vector (ka, kb)
- Geometric interpretation where a vector is an arrow:
- k = – 1 : reverse the direction of an arrow
- k = 2 : double the length of an arrow; same as adding a vector to itself

- One type of vector multiplication is called the dotproduct, it yields a scalar (a value)
- Example: (a, b, c) (d, e, f) = ad + be + ef
- It works in all dimensions
- The dot product rule/equality for vectors u and v:u v = |u||v| cos
- Perpendicular vectors have a dot product 0

- Another type of multiplication is the cross product, denoted by
- It applies only to two vectors in 3D and yields a vector in 3D
- the result is normal to the input vectors
- if the input vectors are parallel, we get the null vector (0, 0, 0)

- The length of the result vector of the cross product is related to the lengths of the input vectors and their angle|a b| = |a||b| sin In words: the length of the resulta b is the area of the parallelogram with a and bas sides

- Other terms of importance:
- linear independence
- spanning a (sub)space
- basis
- orthogonal basis
- orthonormal basis

- Matrices are grids of values; an m-by-n (mn) matrix consists of m rows and n columns
- An mnmatrix represents a linear transformation from m-space to n-space, but it could represent many other things

- A linear transformation:
- maps any point/vector to exactly one point/vector
- maps the origin/null vector to the origin/null vector
- preserves straightness: mapping a line segment (its points) yields a line segment (its points), which can degenerate to a single pointExample:

=

point or vector

mirror in y-axis

shear the x-coordinate

scale x and y by 1.5

rotate by = /6 radians

- Matrix addition: entry-wise
- Multiplication with scalar: entry-wise
- Multiplication of two matrices A and B:
- #columns of A must match #rows of B
- not commutative
- AB represents the lineartransformation whereB is applied first and Ais applied second

- Other terms of importance:
- null matrix (mn), identity matrix (nn)
- rank of a matrix: number of independent rows (or columns)
- determinant: converts a square matrix to a scalarGeometric interpretation: tells something about the area/volume enlargement (2D/3D) of a matrixDet = 2 (in 2D): a transformed triangle has twice the areaDet = 0: the transformation is a projection
- matrix inversion: represents the transformation that is the reverse of what the matrix did
- Gaussian elimination: process (algorithm) that allows us to invert a matrix, or solve a set of linear equations

- A 3x3 matrix can represent any linear transformation from 3-space to 3-space, but no other transformation
- The most important missing transformation is translation (which never maps the origin to the origin so it cannot be a linear transformation)

- Combinations of linear transformations and translations (one applied after the other) are called affine transformations
- Using homogeneous coordinates, we can use a 4x4 matrix to represent all 3-dim affine transformations (generally: (d+1)x(d+1) matrix for d-dim affine tr.) the homogeneous coordinates of the point (a, b, c) are simply (a, b, c, 1)

- The matrix for translation by the vector (a, b, c) using homogeneous coordinates is:Just apply this matrix to the origin = (0, 0, 0, 1) and see where it ends up: (a, b, c, 1)

- It is possible to define and use vectors of points:( (a, b), (c, d), (e,f) ) instead of vectors of scalars
- We can add two of these because vector addition is naturally defined
- We can also multiply such a thing by a scalar( (a, b), (c, d), (e,f) ) + ( (g, h), (i, j), (k,l) ) = ( (a, b)+(g, h), (c, d)+(i, j), (e,f)+(k,l) ) =( (a+g, b+h), (c+i, d+j), (e+k, f+l) ) 3 ( (a, b), (c, d), (e,f) ) = ( 3(a, b), 3(c, d), 3(e,f) ) = ( (3a, 3b), (3c, 3d), (3e, 3f) )

- We can not add such a thing and a normal 3D vector because we cannot add a scalar and a vector/point( (a, b), (c, d), (e,f) ) + ( g, h, i) = undefined

- We can even apply (scalar) matrices to these things:

=

=

This works be cause we know how to add points and multiply scalars and points

- Are the vectors (2, 4, 5), (5, – 1, 1), and (1, –9, –9) linearly independent?
- Multiply
- Find the matrix for the 3D affine transformation: mirror in the plane y – z = 3
- Does the property that the determinant of a square matrix represents the change factor in area/volume of a shape also hold for matrices using homogeneous coordinates? Explain why or why not

- Let S be the collection of all strings. Define
- addition of two strings as their concatenation
- multiplication of a string with a nonnegative integer by repeating the string as often as the value of the integer
Compute: