Recap of linear algebra: vectors, matrices, transformations, … Background knowledge for 3DM Marc van Kreveld

Vectors, points 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 once the point (3, 2)

Vectors, length

Vector addition 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

Scalars, vectors, multiplication 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

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

Vector multiplication 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)

Vector multiplication 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 result a  b is the area of the parallelogram with a and b as sides

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

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

Matrices 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 point Example: point or vector

Matrices mirror in y-axis shear the x-coordinate

Matrices scale x and y by 1.5 rotate by  =  /6 radians

Matrices 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 linear transformation where B is applied first and A is applied second

Matrices Other terms of importance: – null matrix (m  n), identity matrix (n  n) – rank of a matrix: number of independent rows (or columns) – determinant: converts a square matrix to a scalar Geometric interpretation: tells something about the area/volume enlargement (2D/3D) of a matrix Det = 2 (in 2D): a transformed triangle has twice the area Det = 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

Translations and matrices 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)

Homogeneous coordinates 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)

Homogeneous coordinates 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)

Vectors of points 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) )

Vectors of points 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

Vectors of points We can even apply (scalar) matrices to these things: This works be cause we know how to add points and multiply scalars and points

Questions

5.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: