Null Space ಅನ್ನು ಹೇಗೆ ಲೆಕ್ಕ ಹಾಕುವುದು
Null Space ಎಂದರೇನು?
The null space (kernel) of a matrix A is the set of all vectors x such that Ax = 0. The nullity is the dimension of the null space. Understanding null space is essential in solving linear systems and linear algebra.
ಸೂತ್ರ
Null space of A: all vectors x where Ax = 0
- A
- matrix
- x
- vector in null space
- Null(A)
- null space (kernel) of A
ಹಂತ-ಹಂತದ ಮಾರ್ಗದರ್ಶಿ
- 1Solve Ax = 0 using row reduction (RREF)
- 2Free variables correspond to null space dimensions
- 3Rank-nullity theorem: rank + nullity = n (columns)
- 4Null space is always a subspace containing the zero vector
Worked Examples
ಇನ್ಪುಟ್
[[1,2,3],[4,5,6]] × x = 0
ಫಲಿತಾಂಶ
Null space has dimension 1; one free variable
Frequently Asked Questions
Is the zero vector always in the null space?
Yes, A × 0 = 0 always. So the null space is never empty.
What is the relationship between null space and rank?
Rank-nullity theorem: rank(A) + nullity(A) = number of columns. (Nullity = dimension of null space.)
Can the null space be trivial (only zero)?
Yes, if A has full column rank. If A has dependent columns, null space is larger.
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