104_Linear Algebra A gentle introduction

Scalar Product: av , Change only the length “scaling”, but keep direction fixed.. Sneak peek: matrix operation Av can change length,... ❑ Matrix multiplication AB: apply transformatio

Shivkumar Kalyanaraman

Linear Algebra

A gentle introduction

Linear Algebra has become as basic and as applicable

as calculus, and fortunately it is easier.

Gilbert Strang, MIT

What is a Vector ?

❑ Think of a vector as a directed line segment in N-dimensions! (has

“length” and “direction”)

❑ Basic idea: convert geometry in higher dimensions into algebra!

❑ Once you define a “nice” basis along

each dimension: x-, y-, z-axis …

❑ Vector becomes a 1 x N matrix!

❑ v = [a b c]T

❑ Geometry starts to become linear

algebra on vectors like v!

a

y

v

Shivkumar Kalyanaraman

A

B A

B

C

A+B = C (use the head-to-tail method

to combine vectors)

A+B

Scalar Product: av

) ,

( )

Change only the length (“scaling”), but keep direction fixed.

Sneak peek: matrix operation (Av) can change length,

B A B

Inner (dot) Product: v.w or w T v

v

w

α

2 2

1 1 2

1 2

(

The inner product is a

The inner product is a SCALAR! SCALAR!

).(

, (

w v

w

v = 0 ⇔ ⊥

Shivkumar Kalyanaraman

Bases & Orthonormal Bases

❑ Basis (or axes): frame of reference

vs

Basis: a space is totally defined by a set of vectors – any point is a linear

combination of the basis

Ortho-Normal: orthogonal + normal

z x

y x

[ ] [ ] [ ]T

T T

z y x

1 0 0

0 1 0

0 0 1

=

=

=

b a

rows

columns

Shivkumar Kalyanaraman

Basic Matrix Operations

❑ Addition, Subtraction, Multiplication: creating new matrices (or functions)

g c

f b

e

a h

g

f

e d

g c

f b

e

a h

g

f

e d

dg ce

bh af

bg

ae h

g

f

e d

c

b

a

Just add elements

Just subtract elements

Multiply each row

by each column

Matrix Times Matrix

N M

31

23 22

21

13 12

11

33 32

31

23 22

21

13 12

11

33 32

31

23 22

21

13 12

11

n n

n

n n

n

n n

n

m m

m

m m

m

m m

m

l l

l

l l

l

l l

l

32 13 22

12 12

11

Shivkumar Kalyanaraman

Multiplication

❑ Is AB = BA? Maybe, but maybe not!

❑ Matrix multiplication AB: apply transformation B first, and then again transform using A!

❑ Heads up: multiplication is NOT commutative!

❑ Note: If A and B both represent either pure “rotation” or “scaling” they can be interchanged (i.e AB = BA)

g

f

e d

c

b

a h g

f e

Matrix operating on vectors

❑ Matrix is like a function that transforms the vectors on a plane

❑ Matrix operating on a general point => transforms x- and y-components

❑ System of linear equations: matrix is just the bunch of coeffs !

Shivkumar Kalyanaraman

Direction Vector Dot Matrix

c b

Matrices: Scaling, Rotation, Identity

❑ Pure scaling, no rotation => “diagonal matrix” (note: x-, y-axes could be scaled differently!)

❑ Pure rotation, no stretching => “orthogonal matrix” O

❑ Identity (“do nothing”) matrix = unit scaling, no rotation!

Shivkumar Kalyanaraman

2D Translation

t P

P’

t P

P ' = ( x + tx, y + ty) = +

Pyty

P’

t

0 1

0

0 0

1

I

Shivkumar Kalyanaraman

Determinant of a Matrix

❑ Used for inversion

❑ If det(A) = 0, then A has no inverse

b

d bc

ad

A 1 1

http://www.euclideanspace.com/maths/algebra/matrix/functi

ons/inverse/threeD/index.htm

Projection: Using Inner Products (I)

Shivkumar Kalyanaraman

Homogeneous Coordinates

❑ Represent coordinates as (x,y,h)

❑ Actual coordinates drawn will be (x/h,y/h)

Shivkumar Kalyanaraman

Homogeneous Transformations

1 0

0

0

1

1 1

0 0

0 1

3 3

3 3

2 2

2 2

1 1

1 1

3 3

3 3

2 2

2 2

1 1

1 1

+ +

+

=

+ +

+

=

′

+ +

+

=

′

+ +

x

z y

x z

z y

x y

z y

x x

z y x

z

y

x

v v

v

d v

c v

b v

a v

d v

c v

b v

a v

d v

c v

b v

a v

v v

v

d c

b a

d c

b a

d c

b a

v

v

v

v M v

Order of Transformations

❑ Note that matrix on the right is the first applied

❑ Mathematically, the following are equivalent

p’ = ABCp = A(B(Cp))

❑ Note many references use column matrices to represent points In terms of column matrices

p’T = pTCTBTAT

Shivkumar KalyanaramanRensselaer Polytechnic Institute

Vectors: Cross Product

perpendicular to A and B

A and B

in a right-handed coordinate system

) sin( θ

B A

B

B A×B

Shivkumar Kalyanaraman

MAGNITUDE OF THE CROSS

PRODUCT

DIRECTION OF THE CROSS

PRODUCT

❑ The right hand rule determines the direction of the cross product

Shivkumar Kalyanaraman

For more details

❑ Prof Gilbert Strang’s course videos: