104_Linear Algebra A gentle introduction

Linear Algebra A gentle introduction Linear Algebra has become as basic and as applicable as calculus, and fortunately it is easier Gilbert Strang, MIT Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 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 x N matrix! v = [a b c]T Geometry starts to become linear algebra on vectors like v! a     v = b   c  y v x Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” Vector Addition: A+B A+B A A+B = C (use the head-to-tail method to combine vectors) B C B A Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” Scalar Product: av av = a ( x1 , x2 ) = (ax1 , ax2 ) av v Change only the length (“scaling”), but keep direction fixed Sneak peek: matrix operation (Av) can change length, direction and also dimensionality! Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” Vectors: Dot Product d  A ×B = AT B = [ a b c ]  e  = ad + be + cf  f  The magnitude is the dot product of a vector with itself A = AT A = aa + bb + cc A ⋅ B = A B cos(θ ) Think of the dot product as a matrix multiplication The dot product is also related to the angle between the two vectors Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” Inner (dot) Product: v.w or wTv v α w v.w = ( x1 , x2 ).( y1 , y2 ) = x1 y1 + x2 y2 The inner product is a SCALAR! v.w = ( x1 , x2 ).( y1 , y2 ) =|| v || ⋅ || w || cos α v.w = ⇔ v ⊥ w If vectors v, w are “columns”, then dot product is wTv Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 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 x = [1 0] T T [Sneak peek: y = [ 0] Orthogonal: dot product is zero T [ ] z = 0 Normal: magnitude is one ] x⋅ y = x⋅z = y⋅z = Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” What is a Matrix? ❑ A matrix is a set of elements, organized into rows and columns rows columns a b  c d    Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” Basic Matrix Operations ❑ Addition, Subtraction, Multiplication: creating new matrices (or functions) a b   e c d  +  g    f  a + e b + f  =  h  c + g d + h  a b   e c d  −  g    f  a − e b − f  =  h  c − g d − h  a b   e c d   g   f  ae + bg =  h  ce + dg af + bh cf + dh  Just add elements Just subtract elements Multiply each row by each column Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” Matrix Times Matrix L = M⋅N l11 l12 l l 21 22  l31 l32 l13   m11 l23  = m21 l33   m31 m12 m22 m32 m13   n11 m23  ⋅ n21 m33   n31 n12 n22 n32 n13  n23  n33  l12 = m11n12 + m12 n22 + m13n32 Shivkumar Kalyanaraman Rensselaer Polytechnic Institute 10 : “shiv rpi” Scaling P’ P a.k.a: dilation (r >1), contraction (r