6.837 Linear Algebra Review : 6.837 Linear Algebra Review Patrick Nichols
Thursday, September 18, 2003
Overview : Overview Basic matrix operations (+, -, *)
Cross and dot products
Determinants and inverses
Homogeneous coordinates
Orthonormal basis
Additional Resources : Additional Resources 18.06 Text Book
6.837 Text Book
6.837-staff@graphics.lcs.mit.edu
Check the course website for a copy of these notes
What is a Matrix? : What is a Matrix? A matrix is a set of elements, organized into rows and columns rows columns
Basic Operations : Basic Operations Addition, Subtraction, Multiplication Just add elements Just subtract elements Multiply each row by each column
Multiplication : Multiplication Is AB = BA? Maybe, but maybe not!
Heads up: multiplication is NOT commutative!
Vector Operations : Vector Operations Vector: 1 x N matrix
Interpretation: a line in N dimensional space
Dot Product, Cross Product, and Magnitude defined on vectors only
x y v
Vector Interpretation : Vector Interpretation Think of a vector as a line in 2D or 3D
Think of a matrix as a transformation on a line or set of lines V V’
Vectors: Dot Product : Vectors: Dot Product Interpretation: the dot product measures to what degree two vectors are aligned A B A B C A+B = C
(use the head-to-tail method to combine vectors)
Vectors: Dot Product : Vectors: Dot Product Think of the dot product as a matrix multiplication The magnitude is the dot product of a vector with itself The dot product is also related to the angle between the two vectors – but it doesn’t tell us the angle
Vectors: Cross Product : Vectors: Cross Product The cross product of vectors A and B is a vector C which is perpendicular to A and B
The magnitude of C is proportional to the cosine of the angle between A and B
The direction of C follows the right hand rule – this why we call it a “right-handed coordinate system”
Inverse of a Matrix : Inverse of a Matrix Identity matrix: AI = A
Some matrices have an inverse, such that: AA-1 = I
Inversion is tricky: (ABC)-1 = C-1B-1A-1
Derived from non-commutativity property
Determinant of a Matrix : Determinant of a Matrix Used for inversion
If det(A) = 0, then A has no inverse
Can be found using factorials, pivots, and cofactors!
Lots of interpretations – for more info, take 18.06
Determinant of a Matrix : Determinant of a Matrix Sum from left to right
Subtract from right to left
Note: N! terms
Inverse of a Matrix : Inverse of a Matrix Append the identity matrix to A
Subtract multiples of the other rows from the first row to reduce the diagonal element to 1
Transform the identity matrix as you go
When the original matrix is the identity, the identity has become the inverse!
Homogeneous Matrices : Homogeneous Matrices Problem: how to include translations in transformations (and do perspective transforms)
Solution: add an extra dimension
Orthonormal Basis : Orthonormal Basis Basis: a space is totally defined by a set of vectors – any point is a linear combination of the basis
Ortho-Normal: orthogonal + normal
Orthogonal: dot product is zero
Normal: magnitude is one
Example: X, Y, Z (but don’t have to be!)
Orthonormal Basis : Orthonormal Basis X, Y, Z is an orthonormal basis. We can describe any 3D point as a linear combination of these vectors.
How do we express any point as a combination of a new basis U, V, N, given X, Y, Z?
Orthonormal Basis : Orthonormal Basis (not an actual formula – just a way of thinking about it) To change a point from one coordinate system to another, compute the dot product of each coordinate row with each of the basis vectors.
Questions? : Questions? ?