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CM20144 Applications I Mathematics for
Applications Mark Wood cspmaw_at_cs.bath.ac.uk http
//www.cs.bath.ac.uk/cspmaw
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Todays Tutorial

• Line and Plane Equations
• Linear Dependence and Orthogonality
• Linear Transformations
• Vector Subspaces
• Return of Coursework 1 and Questions
• General Questions for Coursework 2

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Finding (Parametric) Line Equations

• Vector equation of a line from two points
• Vector between two points u, v is v u
• So, first get to the line from 0 by adding u
• Then add some multiple, t, of (v u)
• So equation is L u t(v u), where t ? R
• Every point on the line corresponds to some
  value of t
• For example, in 3D where L (x, y, z)
• x u1 t(v1 u1)
• y u2 t(v2 u2)
• z u3 t(v3 u3)

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Finding Plane Equations

• Plane equation from point and normal
• Normal to a plane in d-dimensions (n1, , nd)
• Each element of the normal is equal to a
  coefficient in the plane equation
• For example, in 3D where the normal is (n1, n2,
  n3)
• n1x n2y n3z c is the plane equation
• How to find c?
• Need any point on the plane (x0, y0, z0)
• Substitute into plane equation x x0, y y0,
  z z0
• Then can evaluate c

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Hint for Coursework

• To find the point of intersection between a line
  and plane
• Find the parametric line equation
• Find the plane equation
• Both must be satisfied
• so substitute the line equation into the plane
  equation and solve for t
• Substitute t back into the line equation to find
  the point of intersection
• Now you just have to code it

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Definitions

• Linear Dependence
• Consider c1v1 cmvm 0
• If this equation can be satisfied with c1, , cm
  not all zero, then v1, , vm are linearly
  dependent
• If they must all be zero, then the vectors are
  linearly independent
• Orthogonal Set
• Consider v1, , vm
• This is an orthogonal set if vi and vj are
  orthogonal for all i ? j
• Equivalently, vi . vj 0 for all i ? j

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Examples
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Theorem

• An orthogonal set of nonzero vectors in a vector
  space is linearly independent
• Theorem 5.18 in book, p. 258
• Coursework hint you may want to use this
  theorem and / or the proof

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Definitions

• Linear Transformation
• T(u v) T(u) T(v)
• T(cu) cT(u), c ? R
• eg. T(u) Au (where A is a matrix) is always
  linear
• Vector Subspace
• Start with some vector space V
• Suppose X is a subset of V
• Then X is a subspace of V if
• x, y ? X ? x y ? X (closed under addition)
• x ? X, c ? R ? cx ? X (closed under scalar multn)

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Examples
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Final Hints

• Use these definitions wisely!
• For general advice, look at the slides for
  Tutorial 4 (up on my web page)
• Key things to remember
• Write down definitions you may get some marks
• For Q. 5, hand in a print out of a single Maple
  worksheet containing your function code along
  with subsequent test input, test output and any
  comments you wish to add
• Good luck