silemus.
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  1. 23×33\times 3 determinant expansion
  2. 10Eigenvalues of a 3×33\times 3 matrix
  3. 12Eigenvector for eigenvalue 3
  4. 14Inverse of a 2×22\times 2 matrix
  5. 17Solving a 3×33\times 3 linear system
  6. 20System with no solution
  7. 21Singular matrix, det=0
  8. 22Eigenvalue zero of a matrix
  9. 27Null space of a matrix
  10. 37Volume via scalar triple product
  11. 43Characteristic polynomial, 2×22\times 2
  12. 47Orthogonal matrix check
  13. 48Gram-Schmidt second vector
  14. 49Least-squares line fit
  15. 50Coordinates in a new basis
  16. 51Distance point to plane

27Null space of a matrix

Find a vector spanning the null space of A=[1111]\displaystyle A = \begin{bmatrix}1 & 1\\1 & 1\end{bmatrix}, with first entry 1\displaystyle 1.

Solve Ax = 0: since both rows of A are [1,1], the equation is x1 + x2 = 0, so x2 = −x1. Setting x1 = 1 gives the null space vector (1, −1). Check: 1(1)+1(−1) = 0 in both rows.

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