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Recap of linear algebra: vectors, matrics, transformations, … Background knowledge for 3DM Marc van Kreveld 1 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 the point (3, 2) 2 Vectors, length 3 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 4 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 5 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 + 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 6 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) 7 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 8 Vectors • Other terms of importance: – – – – – linear independence spanning a (sub)space basis orthogonal basis orthonormal basis 9 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 10 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 11 Matrices mirror in y-axis shear the x-coordinate 12 Matrices scale x and y by 1.5 rotate by = /6 radians 13 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 14 Matrices • Other terms of importance: – identity matrix – rank of a matrix – 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 15 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) 16 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) 17 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) 18 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) ) 19 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 20 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 21 Questions 22 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: 23