Tutorial2021Proposed solution included

MATS 2104 Linear Methods Tutorial 2021

MATS 2104: Homework & Tutorial Sheet 4, First Semester 2021, University of Bamenda, Faculty of Science, Department of Mathematics and Computer Science: all three uploaded source files for this course were duplicate scans of this single 14-question tutorial sheet on inner product spaces (no exam paper was present in the uploaded material). Covers: inner-product-axiom proofs for sums/positive scalar multiples; expanding inner-product/norm expressions with complex scalars; orthogonal sets under scalar multiples; finding scalars making a vector orthogonal to one or two orthogonal vectors; the triangle-inequality-type norm bound and its real-space refinement; unit-vector equality/negation in terms of the inner product; the Cauchy-Schwarz equality case implying linear dependence; the complex polarization identity; real-space norm/orthogonality equivalences and their complex counterparts; orthogonal-complement identities for sums and intersections of subspaces; the direct-sum decomposition W+W^perp; a projection-map characterization of V=p(V)+W^perp; and W subset W^(perp perp).

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University of Bamenda

Faculty of Science - FS

Mathematics & Computer Science - MCS

MATS21042021

Linear Methods

Tutorial · First Semester · Level 200

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Let v and w ̸ = 0 be vectors in an inner product space. Find the scalar c such that v + cw is orthogonal to w.
Let v, w1, w2 be vectors in an inner product space such that w1, w2 are nonzero orthogonal vectors. Find the scalars c1, c2 such that v + c1w1 + c2w2 is orthogonal to both w1 and w2.

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