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Vectors and Spaces

Vectors​

  • Vector = Magnitude + Direction

Ex - 5mph is a scaler quantity, because it doesn't tell the direction in which object is moving.

Ex - 5mph East is a vector quantity (and we will not call this as speed, we will call this as velocity, therefore velocity is a vector quantity)

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  • Real Coordinate Space - Can be any dimensions, all possible real-valued ordered 2-tuple for a 2 dimensional real coordinate space.

  • Adding Vectors algebraically and graphically

  • Multiplying vector by a scaler ( change its magnitude, but scale it only on the same dimension, colinear )

  • Unit Vectors

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  • Parametric representations of the line

Linear combinations and spans​

span(v1 + v2 + v3 + ... + vn) = {c1v1 + c2v2 + c3v3 + ... + cnvn | ci belongs to set of Real numbers and 1 <= i <= n}

Linear dependence and independence​

  • Linear dependent - Means one vector in the set can be represented as other vectors in the set.

  • Linear Independent - If we cannot scale using any scalar one vector to the other vector in the set, than both are linearly independent of each other

C1 and C2 must be equal to 0 for R2

Subspaces and the basis for a subspace​

  • Must contain 0 vector
  • Closure under scalar Multiplication - In order to be in a subspace, any vector multiplied by a real scalar must also be in the subspace
  • Closure under Addition - If we add two vectors belonging to the same subspace, than the addition of both the vectors must also belong to the same subspace.
  • Span of n vectors is a valid subspace of Rn
  • S is a basis of V, if something is a basis for a set, that means that, if you take the span of those vectors, you can get to any of the vectors in that subspace and that those vectors are linearly independent.
    • Span (s) = R2
    • Must be Linearly Independent
    • Standard Basis = T = {[1 0] , [0 1]}

Vector dot and cross products​

  • Dot product
  • Vector dot product is commutative, V.W = W.V
  • Vector dot product is distributive, (V + W).X = (V.X + W.X)
  • Vector dot product is associative ( (c.V).W = c.(V.W) )
  • Length of vector X = ||X||
  • ||X||^2^ = X.X
  • Cauchy -Schwarz inequality
  - |X.Y|<= ||X||.||Y||
  - |X.Y| = ||X||.||Y|| only when X and Y are colinear i.e. X = c.Y
  • Vector Triangle Inequality
  • Angles between Vectors
    • (A.B) = ||A||*||B||*cos Θ
    • If A and B are perpendicular than there dot product is 0, since cos 90 = 0
    • If A.B = 0 (dot product of vector A and dot product of vector B is equal to 0) and A and B are non zero vectors than A and B are perpendicular to each other.
    • But if only A.B = 0 satisfies, than A and B are orthogonal
    • All perpendicular vectors are Orthogonal
    • Zero vector is orthogonal to everything else, even to itself

Matrices for solving systems by elimination​

Null space and column space​

References​

https://www.khanacademy.org/math/linear-algebra/vectors-and-spaces