6.837 Linear Algebra Review Rob Jagnow Monday, September 20, 2004

6.837 Linear Algebra Review6.837 Linear Algebra Review

6.837 Linear Algebra 6.837 Linear Algebra ReviewReview

Rob JagnowMonday, September 20, 2004


OverviewOverview

• Basic matrix operations (+, -, *)• Cross and dot products• Determinants and inverses• Homogeneous coordinates• Orthonormal basis


Additional ResourcesAdditional Resources

• 18.06 Text Book• 6.837 Text Book• [email protected]• 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

1110

0100

aa

aa

n columns

m rows

m×n matrix


Basic OperationsBasic Operations

• Transpose: Swap rows with columns

ihg

fed

cba

M

ifc

heb

gda

M T

z

y

x

V zyxV T



• Addition and Subtraction

hdgc

fbea

hg

fe

dc

ba

hdgc

fbea

hg

fe

dc

ba

Just add elements

Just subtract elements

A

B

A

BA+B

AA

-B

-BA-B

A

A

B



• Multiplication

dhcfdgce

bhafbgae

hg

fe

dc

ba Multiply each row by each column

An m×n can be multiplied by an n×p matrix to yield an m×p result


MultiplicationMultiplication

• Is AB = BA? Maybe, but maybe not!

• Heads up: multiplication is NOT commutative!

......

...bgae

hg

fe

dc

ba

......

...fcea

dc

ba

hg

fe


Vector OperationsVector Operations

• Vector: n×1 matrix• Interpretation:

a point or line in n-dimensional space

• Dot Product, Cross Product, and Magnitude defined on vectors only

c

b

a

v

x

y

v


Vector InterpretationVector 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

y

x

dc

ba

y

x

'

'V

V’


Vectors: Dot ProductVectors: Dot Product

• Interpretation: the dot product measures to what degree two vectors are aligned

A

B

A

B A=B

If A and B have length 1…

A·B = 0 A·B = 1 A·B = cos θ

θ


Vectors: Dot ProductVectors: Dot Product

cfbead

f

e

d

cbaABBA T

ccbbaaAAA T 2

)cos(BABA

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


Vectors: Cross ProductVectors: 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 sin of the angle between A and B

• The direction of C follows the right hand rule if we are working in a right-handed coordinate system

)sin(BABA B

A

A×B


Vectors: Cross ProductVectors: Cross Product

The cross-product can be computed as a specially constructed determinant

A

B

A×B

xyyx

zxxz

yzzy

zyx

zyx

baba

baba

baba

bbb

aaa

kji

BA

ˆˆˆ


Inverse of a MatrixInverse 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

100

010

001

I


Determinant of a MatrixDeterminant 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

dc

baA

bcadA )det(

ac

bd

bcadA

11


Determinant of a MatrixDeterminant of a Matrix

cegbdiafhcdhbfgaei

ihg

fed

cba

ihg

fed

cba

ihg

fed

cba

ihg

fed

cbaFor a 3×3 matrix:Sum from left to rightSubtract from right to left

Note: In the general case, the determinant has n! terms


Inverse of a MatrixInverse of a Matrix

100

010

001

ihg

fed

cba

1. Append the identity matrix to A

2. Subtract multiples of the other rows from the first row to reduce the diagonal element to 1

3. Transform the identity matrix as you go

4. When the original matrix is the identity, the identity has become the inverse!


Homogeneous MatricesHomogeneous Matrices

• Problem: how to include translations in transformations (and do perspective transforms)

• Solution: add an extra dimension

110001

'

'

'

222120

121110

020100

z

y

x

taaa

taaa

taaa

z

y

x

z

y

x


Orthonormal BasisOrthonormal Basis

• Basis: a space is totally defined by a set of vectors – any point is a linear combination of the basis

• Orthogonal: dot product is zero• Normal: magnitude is one• Orthonormal: orthogonal + normal• Most common Example: zyx ˆ,ˆ,ˆ


Change of Orthonormal Change of Orthonormal BasisBasis

• Given: coordinate frames

xyz and uvnpoint p = (px, py, pz)

• Find:p = (pu, pv, pn)

p

x

y

z

v

u

x

yv

u

n

px

y

uv



(x . u) u(y . u) u(z . u) u

(x . v) v(y . v) v(z . v) v

(x . n) n(y . n) n(z . n) n

xyz

===

+++

+++

x

y

z

v

u

n

x . u

x . v

y . v

y . u

x

yv

u



(x . u) u(y . u) u(z . u) u

(x . v) v(y . v) v(z . v) v

(x . n) n(y . n) n(z . n) n

xyz

===

+++

+++

p = (px, py, pz) = px x + py y + pz z

(x . u) u(y . u) u(z . u) u

(x . v) v(y . v) v(z . v) v

(x . n) n(y . n) n(z . n) n

+++

+++

px [py [pz [

] +] +]

Substitute into equation for p:

p =



(x . u) u(y . u) u(z . u) u

(x . v) v(y . v) v(z . v) v

(x . n) n(y . n) n(z . n) n

p = +++

+++

px [py [pz [

] +] +]

px (x . u)px (x . v)px (x . n)

py (y . u)py (y . v)py (y . n)

pz (z . u)pz (z . v)pz (z . n)

+++

+++

[[[

] u +] v +] n

Rewrite:

p =




pu

pv

pn

===

+++

+++



p = px (x . u)px (x . v)px (x . n)



+++

+++

[[[

] u +] v +] n

Expressed in uvn basis:

p = (pu, pv, pn) = pu u + pv v + pn n



pu

pv

pn

=

px

py

pz

ux

vx

nx

uy

vy

ny

uz

vz

nz

In matrix form:

ux = x . uwhere:

etc.

uy = y . u


pu

pv

pn

===

+++

+++





What's M-1, the inverse?

ux = x . u = u . x = xupu

pv

pn

=

px

py

pz

xu

yu

zu

xv

yv

zv

xn

yn

zn

= M

px

py

pz

M-1 = MT

pu

pv

pn

=

px

py

pz

ux

vx

nx

uy

vy

ny

uz

vz

nz


CaveatsCaveats

• Right-handed vs. left-handed coordinate systems– OpenGL is right-handed

• Row-major vs. column-major matrix storage.– matrix.h uses row-major order– OpenGL uses column-major order

15141312

111098

7654

3210

151173

141062

13951

12840

row-major column-major


Questions?Questions?

?

Documents

6.837 Linear Algebra Review Rob Jagnow Monday, September 20, 2004