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

50Coordinates in a new basis

Find the coordinates of (5,1)\displaystyle (5, 1) in the basis {(1,1),(1,−1)}\displaystyle \{(1, 1), (1, -1)\}.

Write (5,1) as c1(1,1) + c2(1,−1), which splits into c1+c2=5 and c1−c2=1. Adding gives c1=3, then c2=2, so the coordinates in this basis are (3, 2). Check: 3(1,1)+2(1,−1) = (5,1).

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