Inverse matrices, column space and null space

What to learn

  • What is Inverse matrices
  • What is Column Space
  • What is Rank
  • What is null space

    Description

  • Why linear algebra is useful?
    • It lets us solve certain systems of equations
  • System of equations
    • we have list of variables and equations relating them

    • Linear System of equations example

      \[\begin{matrix} 2x+5y+3z = -3\\ 4x+0y+8z = 0\\ 1x+3y+0z=2 \end{matrix}\] \[\rightarrow \begin{bmatrix}2 & 5 & 3\\ 4 & 0 & 8\\ 1 & 3 & 0\end{bmatrix}\begin{bmatrix} x\\ y\\ z \end{bmatrix} = \begin{bmatrix} 3\\ 0\\ 1 \end{bmatrix}\] \[\rightarrow A\vec{x} = \vec{v}\]
    • It means find vector x using transformed vector v by linear transformation A
    • How we find solution of this equation is depands on whether the transformation A squish space to lower dimension or not.(first step we have to find) → case determinant is zero or not
      • Nonzero case
        • Linear transformation is one-to-one correspondence
          • Only one vector is corresponded to Only one transformated vector.
        • Only one solution in \(x = A^{-1} v\) always exists. and we can call \(A^{-1}\) as the inverse of A
        • \(A^{-1} * A\) equals the transformation that does nothing. → identity transformation
        • \(det(A) ≠ 0\), A doesn’t squish the space
      • Zero case
        • In \(det(A) = 0\) case, A squish the space to lower dimension. So we can’t find Inverse of A. we can’t unsquish line to plane using function.
          • Because single vector has to become multiple vector to broaden dimension.
        • But we can find the infinite solutions if transformation squish a plane to line while x and v are same direction(linearly dependant) and others do not exist solution.
      • Rank
        • Number of dimensions in the output
          • we want to know after transformation, how many dimensions are squished.
          • When the output of transformation is line, means one dimensional, then we can say transformation’s rank is 1
          • if All the vectors land on two dimensional plane, then we can say transformation’s rank is 2
        • Max rank of 2x2 transformation is 2
      • Column space
        • Set of all possible outputs Ax
        • Span of columns ↔ Column space
          • Rank is number of dimensions in the column space
          • If column space equals number of column, then we can say it as full rank
      • Zero vector
        • It is always included in the column space
          • Because linear transformation always include fixed origin.
        • For a full rank transformation, only vector to origin itself is the zero vector.
      • Null space = kernel
        • If transformation makes vector to origin, then we can say that set of vectors as Null space
          • If plane is squished to line, then there are lots of vectors in same line become zero vector.
          • If space is squished to plane, then there are also lots of vectors in same line become zero vector.
          • If space is squished to line, then there are lots of vectors in plane vectors become zero vector.
        • \(A \vec{x} = \vec{0}\) → all null space be possible solution.
    • Overview
      • Each system has some kind of linear transformation associated with it. and when that transformation has an inverse, we can use that inverse to solve our system.
      • column space lets us understand when a solution even exists.
      • null space helps us understand what the set of all possible solution can look like.

        Next Step

  • Nonsquare matrices as transformations between dimensions
  • Gaussian elimination and Row echelon form

    References

  • https://www.youtube.com/watch?v=rHLEWRxRGiM&list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab&index=5