Coordinates of u in basis v1, v2, v3, v4 are numbers s1, s2, s3, s4, such as:
u=s1v1+s2v2+s3v3+s4v4
We can write this using matrix:
⎝⎛v11v12v13v14v21v22v23v24v31v32v33v34v41v42v43v44⎠⎞⎝⎛s1s2s3s4⎠⎞=⎝⎛u1u2u3u4⎠⎞
We've got the linear equation. We can solve it using matrix transformations (Gaussian method):
⎝⎛v11v12v13v14v21v22v23v24v31v32v33v34v41v42v43v44∣∣∣∣u1u2u3u4⎠⎞=
=⎝⎛1−12−1−2232120−11001∣∣∣∣−1121⎠⎞
Add first row to the second and fourth, and subtract 2 times first row from third row:
⎝⎛1000−207013−2011−22∣∣∣∣−1040⎠⎞
Swap second and third rows:
⎝⎛1000−27001−2301−212∣∣∣∣−1400⎠⎞
After transformations, we've got this system of equations:
s1−2s2+s3+s4=−17s2−2s3−2s4=43s3+s4=02s4=0
s1−2s2+s3=−17s2−2s3=43s3=0s4=0
s1−2s2=−17s2=4s3=0s4=0
s1−8/7=−1s2=4/7s3=0s4=0
s1=1/7s2=4/7s3=0s4=0
Check the solution. If solution is correct, we'll get (-1, 1, 2, 1):
u=1/7v1+4/7v2=1/7(v1+4v2)==1/7(1−8,−1+8,2+12,−1+8)==1/7(−7,7,14,7)=(−1,1,2,1)
Solution is correct.
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