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Exercise:
abcliste abc Let Ain textGL_nmathbbR. Then exists Qin On and an upper triangular matrix Rin M_ntimes nmathbbR s.t. AQ R. abc Let Ain textGL_nmathbbR. Then exists Qin Un and an upper triangular matrix Rin M_ntimes nmathbbC s.t. AQ R. abcliste

Solution:
Proof. Write A pmatrix vdots & & vdots v_ & hdots & v_n vdots & & vdots pmatrix with v_kin K^n_textcol KmathbbR or mathbbC. Since Ain textGL_nK v_...v_n form a basis for K^n. Apply G-M orthogonalization to v_...v_n and obtain an orthonormal basis e_...e_n for K^n. Recall that forall leq jleq n textSpv_...v_jtextSpe_...e_j. And we also have: v_ langle v_e_rangle e_ v_ langle v_e_rangle e_+langle v_e_rangle e_ &vdots v_j langle v_je_rangle e_+...+langle v_je_jrangle e_j &vdots v_n langle v_ne_rangle e_+...+langle v_ne_nrangle e_n &Rightarrow pmatrix vdots & & vdots & & vdots v_ & hdots & v_j & hdots & v_n vdots & & vdots & & vdots pmatrix pmatrix vdots & & vdots & & vdots e_ & hdots & e_j & hdots & e_n vdots & & vdots & & vdots pmatrix pmatrix langle v_e_rangle & langle v_e_rangle & vdots & langle v_je_rangle & vdots & langle v_ne_rangle & langle v_e_rangle & vdots & langle v_je_rangle & vdots & langle v_ne_rangle vdots & & ddots & vdots & & vdots vdots & vdots & & langle v_je_jrangle & & vdots & & hdots & & & langle v_ne_nrangle pmatrix
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Exercise:
abcliste abc Let Ain textGL_nmathbbR. Then exists Qin On and an upper triangular matrix Rin M_ntimes nmathbbR s.t. AQ R. abc Let Ain textGL_nmathbbR. Then exists Qin Un and an upper triangular matrix Rin M_ntimes nmathbbC s.t. AQ R. abcliste

Solution:
Proof. Write A pmatrix vdots & & vdots v_ & hdots & v_n vdots & & vdots pmatrix with v_kin K^n_textcol KmathbbR or mathbbC. Since Ain textGL_nK v_...v_n form a basis for K^n. Apply G-M orthogonalization to v_...v_n and obtain an orthonormal basis e_...e_n for K^n. Recall that forall leq jleq n textSpv_...v_jtextSpe_...e_j. And we also have: v_ langle v_e_rangle e_ v_ langle v_e_rangle e_+langle v_e_rangle e_ &vdots v_j langle v_je_rangle e_+...+langle v_je_jrangle e_j &vdots v_n langle v_ne_rangle e_+...+langle v_ne_nrangle e_n &Rightarrow pmatrix vdots & & vdots & & vdots v_ & hdots & v_j & hdots & v_n vdots & & vdots & & vdots pmatrix pmatrix vdots & & vdots & & vdots e_ & hdots & e_j & hdots & e_n vdots & & vdots & & vdots pmatrix pmatrix langle v_e_rangle & langle v_e_rangle & vdots & langle v_je_rangle & vdots & langle v_ne_rangle & langle v_e_rangle & vdots & langle v_je_rangle & vdots & langle v_ne_rangle vdots & & ddots & vdots & & vdots vdots & vdots & & langle v_je_jrangle & & vdots & & hdots & & & langle v_ne_nrangle pmatrix
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eth, fs23, lineare algebra, proof
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(3, default)
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Proof
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