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Linear Algebra for Robotics

1. The Big Picture

taught in MIT

1.1. Row Space = $C(A^T)$

linear combination, fill the plane
$$
A=\left[\begin{array}{lll}
1 & 2 & 3 \
4 & 5 & 6
\end{array}\right]
$$
Only 2 rows, so can’t fill the 3D, here we introduce null space

1.2. Null Space = $N(A)$

$$
A v=\left[\begin{array}{l}
0 \
0
\end{array}\right]
$$

In this situation
$$
v=\left[\begin{array}{l}
1 \
-2 \
1
\end{array}\right]
$$
is the null vector, prependicular line

1.3. Column Space = $C(A)$

The column space the whole space

1.4. Left Space =$N(A^T)$

2. Vectors

$$
\boldsymbol { v } = \left( v _ { x } , v _ { y } , v _ { z } \right)
$$

  • length of a vector

  • p-norm: $| v | _ { p } = \left( \sum _ { i = 1 } ^ { n } \left| v _ { i } \right| ^ { p } \right) ^ { 1 / p }$

  • Euclidean length: $p=2$

  • operation

  • dot product
    $$
    \boldsymbol { a } \cdot \boldsymbol { b } = \boldsymbol { b } \cdot \boldsymbol { a } = \boldsymbol { a } ^ { T } \boldsymbol { b } = \boldsymbol { b } ^ { T } \boldsymbol { a } = \sum _ { i = 1 } ^ { n } a _ { i } b _ { i } = | a | _ { 2 } | b | _ { 2 } \cos \theta
    $$

  • cross product
    $$
    \boldsymbol { a } \times \boldsymbol { b } = - \boldsymbol { b } \times \boldsymbol { a } = \operatorname { det } \left( \begin{array} { l l l } { \hat { \boldsymbol { x } } } & { \hat { \boldsymbol { y } } } & { \hat { z } } \ { a _ { 1 } } & { a _ { 2 } } & { a _ { 3 } } \ { b _ { 1 } } & { b _ { 1 } } & { b _ { 3 } } \end{array} \right) = [ a ] _ { \mathbf { x } } \boldsymbol { b } = | a | _ { 2 } | b | _ { 2 } \sin \theta \hat { \boldsymbol { n } }
    $$

3. Matrices

$$
A = \left( \begin{array} { c c c c } { a _ { 1,1 } } & { a _ { 1,2 } } & { \cdots } & { a _ { 1 , n } } \ { a _ { 2,1 } } & { a _ { 2,2 } } & { \cdots } & { a _ { 2 , n } } \ { \vdots } & { \vdots } & { \ddots } & { } \ { a _ { m , 1 } } & { a _ { n , 2 } } & { \cdots } & { a _ { m , n } } \end{array} \right) , A \in \mathbb { R } ^ { m \times n }
$$

3.1. Square Matrices

  • Inverse $A A ^ { - 1 } = A ^ { - 1 } A = I _ { n \times n }$

  • symmetric $A = A ^ { T }$

  • skew-symmetric $A = - A ^ { T }$
    $$
    S = [ v ] _ { \times } = \left( \begin{array} { c c c } { 0 } & { - v _ { z } } & { v _ { y } } \ { v _ { z } } & { 0 } & { - v _ { x } } \ { - v _ { y } } & { v _ { x } } & { 0 } \end{array} \right)
    $$

  • orthogonal: $A ^ { - 1 } = A ^ { T }$

  • The product of two orthogonal matrices of the same size is also an orthogonal matrix

  • Group $O(n)$

  • deteminant $=+1 \to SO(n)$

  • normal: $A ^ { T } A = A A ^ { T }$

  • can be diagonalized by an orthogonal matrix

  • All symmetric, skew-symmetric and orthogonal matrices are normal matrices

  • determinant: factor by which the transformation changes changes volumes in an n-dimensional space;

  • equal to the product of the eigenvalues: $\operatorname { det } ( A ) = \prod _ { i = 1 } ^ { n } \lambda _ { i }$

  • trace: $\operatorname { tr } ( A ) = \sum _ { i = 1 } ^ { n } A _ { i i } = \sum _ { i = 1 } ^ { n } \lambda _ { i }$

  • sum of the diagonal elements

  • sum of the eigenvalues

3.2. Nonsquare Matrices