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. Vectors, points. A vector is an ordered pair, triple, … of (real) numbers, often written as a column

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Recap of linear algebra: vectors, matrics, 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 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 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

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 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 (mn) matrix consists of m rows and n columns

  • An mnmatrix 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 (mn), identity matrix (nn)

    • 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

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


  • 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


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