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https://texercises.com/exercise/gram-schmidt-orthogonalisation-process/
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Exercise:
Let Vlangle rangle be an inner product space. abcliste abc If Ssubseteq V is an orthogonal system then S is linearly indepent. abc If v_...v_n is an orthogonal system and vin textSpv_...v_n and write va_v_+...+a_nv_n a_jin K KmathbbR or mathbbC. Then the coefficients a_j are given by a_j fraclangle vv_j ranglelangle v_j v_j rangle forall leq jleq n. abcliste

Solution:
Proof. abcliste abc Suppose _k^na_kv_k where ngeq v_...v_nin S are distinct and a_...a_nin K. By b a_jfraclangle v_j ranglelangle v_j v_j rangle forall leq jleq n. abc Write va_v_+...+a_nv_n. Let leq jleq n. We have langle vv_j rangle _k^n a_klangle v_kv_j rangle a_jlangle v_jv_j rangle Longrightarrow a_jfraclangle vv_j ranglelangle v_jv_j rangle abcliste
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Exercise:
Let Vlangle rangle be an inner product space. abcliste abc If Ssubseteq V is an orthogonal system then S is linearly indepent. abc If v_...v_n is an orthogonal system and vin textSpv_...v_n and write va_v_+...+a_nv_n a_jin K KmathbbR or mathbbC. Then the coefficients a_j are given by a_j fraclangle vv_j ranglelangle v_j v_j rangle forall leq jleq n. abcliste

Solution:
Proof. abcliste abc Suppose _k^na_kv_k where ngeq v_...v_nin S are distinct and a_...a_nin K. By b a_jfraclangle v_j ranglelangle v_j v_j rangle forall leq jleq n. abc Write va_v_+...+a_nv_n. Let leq jleq n. We have langle vv_j rangle _k^n a_klangle v_kv_j rangle a_jlangle v_jv_j rangle Longrightarrow a_jfraclangle vv_j ranglelangle v_jv_j rangle abcliste
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eth, fs23, gram-schmidt, inner product space, linalg ii, orthogonalisation, proof
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ENG (English)
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