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
- 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
always exists. and we can call as the inverse of A equals the transformation that does nothing. → identity transformation , A doesn’t squish the space
- Linear transformation is one-to-one correspondence
- Zero case
- In
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.
- In
- 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
- Number of dimensions in the output
- 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.
- It is always included in the column space
- 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.
→ all null space be possible solution.
- If transformation makes vector to origin, then we can say that set of vectors as Null space
- Nonzero case
- 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