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

• 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

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)

(𝑎1𝑎2𝑎3)×(𝑏1𝑏2𝑏3)=(𝑎2𝑏3−𝑎3𝑏2𝑎3𝑏1−𝑎1𝑏3𝑎1𝑏2−𝑎2𝑏1)

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 bas 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

(−1 00 1)

(1 1.250 1 )

mirror in y-axis

shear the x-coordinate

Matrices

(1.5 00 1.5)

(c os𝜃 −sin 𝜃sin 𝜃 cos𝜃 )

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 whereB is applied first and Ais 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 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

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)

(1 00 1

0 𝑎0 𝑏

0 00 0

1 𝑐0 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

Questions1. Are the vectors (2, 4, 5), (5, – 1, 1), and (1, –9, –9) linearly

independent?2. Multiply

3. Find the matrix for the 3D affine transformation: mirror in the plane y – z = 3

4. 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

Questions5. 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:

(2 10 2)(b oepf )

(2 1 )(2 10 3)(u sthem )