mat1341notes - MAT1341 Notes By Eric Hua Contents 3.1...

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MAT1341 Notes - By Eric Hua Contents 3.1. Geometric Vectors 3 3.2. Dot Product and Projections 4 3.5. The Cross Product 5 3.3 Lines and Planes 6 1.1 Matrices 8 1.2 Linear Equations 9 1.3. Homogeneous Systems 15 1.4 Matrix multiplication 16 1.5 Matrix Inverses 20 2.1 Cofactor Expansion 22 2.2.1 The Product Theorem 26 2.3.1. A Model of Population Dynamics 27 2.3.2. Eigenvalues and Eigenvectors 27 2.3.3 Diagonalization 29 2.5.1, 2.5.2, 2.5.4: Complex Numbers 33 4.1 Subspaces and Spanning 34 4.2 Linear Independence 37 4.3 Dimension 39 4.4 Rank 41 1
4.5 Orthogonality 43 4.6 Projections and Approximation 48 5.1 Vector Spaces and Subspaces 52 5.2 Independence and Dimension 55 5.3, 4.9.1, 4.9.3: Linear Transformations 57 2
3.1. Geometric Vectors Algebraic representation of vectors: Vectors in R 2 : ⃗v = v = ( a, b ) = [ a b ] = [ a b ] T , zero vector 0 = (0 , 0) . Vectors in R 3 : v = ( a, b, c ) = a b c = [ a b c ] T , zero vector 0 = (0 , 0 , 0) . Length (norm, magnitude) || ( a, b ) || = a 2 + b 2 , || ( a, b, c ) || = a 2 + b 2 + c 2 . Sum: Let ⃗u = ( u 1 , u 2 , u 3 ), ⃗v = ( v 1 , v 2 , v 3 ), then ⃗u + ⃗v ) = ( u 1 + v 1 , u 2 + v 2 , u 3 + v 3 ). Scalar multiple: Let ⃗u = ( u 1 , u 2 , u 3 ), c be a scalar, then c⃗u = ( cu 1 , cu 2 , cu 3 ). Distance: Let ⃗u = ( u 1 , u 2 , u 3 ), ⃗v = ( v 1 , v 2 , v 3 ), then d ( ⃗u,⃗v ) = || ⃗u ⃗v || . unit vector: || ⃗u || = 1 . Example 1. Let ⃗u = (1 , 2 , 2) . Find the unit vector which has the same direction as ⃗u . Properties: Let c , d be scalars. ⃗u + ⃗v = ⃗v + ⃗u ( ⃗u + ⃗v ) + ⃗w = ⃗u + ( ⃗v + ⃗w ) ⃗u + 0 = ⃗u , ⃗u + ( ⃗u ) = 0 ( cd ) ⃗u = c ( d⃗u ) ( c + d ) ⃗u = c⃗u + d⃗u c ( ⃗u + ⃗v ) = c⃗u + c⃗v 1 ⃗u = ⃗u , ( 1) ⃗u = ⃗u , 0 ⃗u = 0 ⃗u//⃗v ⃗v = c⃗u 3
3.2. Dot Product and Projections Dot product: Let ⃗u = ( u 1 , u 2 , u 3 ), ⃗v = ( v 1 , v 2 , v 3 ), then ⃗u · ⃗v = u 1 v 1 + u 2 v 2 + u 3 v 3 . Angle: Let θ be the angle between ⃗u and ⃗v which satisfies 0 θ π , then cos θ = ⃗u · ⃗v || ⃗u || || ⃗v || . Orthogonal: ⃗u ⃗v if ⃗u · ⃗v = 0. Direction angles to the three axis and direction cosines of vectors.: cos α = ⃗u · i || ⃗u || || i || , cos β = ⃗u · j || ⃗u || || j || , cos γ = ⃗u · k || ⃗u || || k || . They satisfy cos 2 α + cos 2 β + cos 2 γ = 1 , ⃗u || ⃗u || = (cos α, cos β, cos γ ) . Projection: The projection of ⃗u onto ⃗v is proj ⃗v ⃗u = ( ⃗u · ⃗v || ⃗v || 2 ) ⃗v, comp ⃗v ⃗u = ⃗u · ⃗v || ⃗v || . Example 2. Let ⃗u = (1 , 2 , 2) , ⃗v = ( 2 , 2 , 1) . Write ⃗u = u 1 + u 2 s.t. u 1 ⃗v , u 2 //⃗v . Example 3. Let ⃗u = (1 , 2 , 2) , ⃗v = ( 2 , 2 , 1) , Find the cosine of the angle between ⃗u and ⃗v . Solution: cos θ = ⃗u · ⃗v || ⃗u || || ⃗v || = 8 9 . Properties: Let c be a scalar. ⃗u · ⃗v = ⃗v · ⃗u ⃗w · ( ⃗u + ⃗v ) = ⃗w · ⃗u + ⃗w · ⃗v c ( ⃗u · ⃗v ) = ( c⃗u ) · ⃗v = ⃗u · ( c⃗v ) ⃗u · 0 = 0 ⃗u · ⃗u = || ⃗u || 2 . 4
3.5. The Cross Product Cross product: Let ⃗u = ( u 1 , u 2 , u 3 ), ⃗v = ( v 1 , v 2 , v 3 ), then ⃗u × ⃗v = ( u 2 v 3 u 3 v 2 , + u 3 v 1 u 1 v 3 , u 1 v 2 u 2 v 1 ) .

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