a r = v b c 1 Dot product (scalar - - PDF document

1 Appendix B: Linear Algebra

• x Rn

Vectors

x y v

⎥ ⎤ ⎢ ⎡a ⎥ ⎥ ⎥ ⎦ ⎢ ⎢ ⎢ ⎣ = c b v r

2 Dot product (scalar product)

B A C A B C

Cross product

• Vector product (cross product)

3 Differentiation of Vectors Linear Independence

4 What is a Matrix?

• A matrix is a set of elements, organized into rows and columns

g

⎥ ⎤ ⎢ ⎡ b a

rows columns

⎥ ⎦ ⎢ ⎣ d c

Basic Operations

• Addition, Subtraction, Multiplication

5 Multiplication

• Is AB = BA? Maybe, but maybe not!
• Heads up: multiplication is NOT commutative!

p p

Matrix Transpose

6 Inverse of a Matrix

• Identity matrix:

AI = A

⎥ ⎥ ⎥ ⎤ ⎢ ⎢ ⎢ ⎡ = 1 1 I ⎥ ⎦ ⎢ ⎣ 1

Determinant of a Matrix

• Used for inversion

⎤ ⎡ b

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

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = d c b a A

7 Determinant of a Matrix b ceg bdi afh cdh bfg aei i h g f e d c b a − − − + + = f e d c b a f e d c b a f e d c b a i h g f e d i h g f e d i h g f e d Inverse of a Matrix

⎥ ⎤ ⎢ ⎡ 1 c b a ⎥ ⎥ ⎥ ⎦ ⎢ ⎢ ⎢ ⎣ + 1 1 i h g f e d

8 Null Space