Basis change --- Introduction ---

This is a classical exercise on vector spaces: one gives a new basis libgcov profiling error:/var/lib/wims/src/Lib/text.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/Lib/dir.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/wims.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/variables.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/var.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/test.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/texmath.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/score.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/rawmath.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/pedia.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/matrix.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/mathml.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/mathfonts.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/log.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/lines.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/insmath.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/html.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/files.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/exec.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/exam.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/evalue.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/config.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/compare.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/cleaning.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/calc.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/auth.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) ℬ of libgcov profiling error:/var/lib/wims/src/Lib/text.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/Lib/dir.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/wims.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/variables.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/var.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/test.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/texmath.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/score.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/rawmath.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/pedia.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/matrix.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/mathml.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/mathfonts.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/log.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/lines.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/insmath.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/html.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/files.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/exec.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/exam.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/evalue.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/config.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/compare.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/cleaning.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/calc.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/auth.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) ℝ n as well as a vector u written under the standard basis of libgcov profiling error:/var/lib/wims/src/Lib/text.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/Lib/dir.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/wims.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/variables.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/var.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/test.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/texmath.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/score.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/rawmath.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/pedia.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/matrix.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/mathml.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/mathfonts.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/log.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/lines.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/insmath.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/html.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/files.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/exec.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/exam.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/evalue.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/config.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/compare.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/cleaning.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/calc.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/auth.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) ℝ n, and you should find the coefficients of u in the basis libgcov profiling error:/var/lib/wims/src/Lib/text.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/Lib/dir.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/wims.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/variables.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/var.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/test.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/texmath.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/score.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/rawmath.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/pedia.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/matrix.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/mathml.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/mathfonts.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/log.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/lines.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/insmath.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/html.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/files.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/exec.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/exam.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/evalue.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/config.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/compare.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/cleaning.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/calc.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/auth.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) ℬ=(v 1,v 2,⋯,v n), that is the real numbers c i libgcov profiling error:/var/lib/wims/src/Lib/text.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/Lib/dir.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/wims.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/variables.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/var.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/test.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/texmath.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/score.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/rawmath.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/pedia.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/matrix.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/mathml.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/mathfonts.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/log.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/lines.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/insmath.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/html.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/files.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/exec.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/exam.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/evalue.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/config.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/compare.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/cleaning.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/calc.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/auth.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) (i=1,…,n) such that :
libgcov profiling error:/var/lib/wims/src/Lib/text.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/Lib/dir.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/wims.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/variables.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/var.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/test.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/texmath.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/score.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/rawmath.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/pedia.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/matrix.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/mathml.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/mathfonts.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/log.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/lines.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/insmath.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/html.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/files.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/exec.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/exam.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/evalue.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/config.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/compare.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/cleaning.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/calc.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) libgcov profiling error:/var/lib/wims/src/auth.gcda:Version mismatch - expected 14.2 (release) (B42*) got 12.2 (release) (B22*) u=∑ i=1 nc iv i.
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