A new approach to linear filtering and prediction problems by Kallenrode

By Kallenrode

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X„ G W, and all scalars a ............. G R. the vector « ,xi + • • • + a„x„ G W. b) Is the converse of the statement in part a true? 5. Let W be a subspace of a vector space V, let y G V, anddefine the set y + W — {x G V | x = y + w for some w G W}. Showthat y + W is a subspace of V if and only if y G W. 6. If W, and W2 are subspaces of a vector space V, is Wx \ W2 ever a subspace of V? Why or why not ? ) 7. a) Show that in V = R ’, each line containing the origin is a subspace. b) Show that the only subspaces of V = R2 are the zero subspace, R 2 itself, and the lines through the origin.

Note that by the choice of j, none of the variables x ,, . , x ^ x can appear explicitly. 5. INTERLUDE ON SOLVING SYSTEMS OF LINEAR EQUATIONS 39 equation and the first equation. Next, we apply an elementary operation of type b and multiply (the new) first equation by 1/a,, to make the leading coefficient equal to 1. 15) @k + 1,/ * 1X j +. ) Now, we apply the induction hypothesis to the system of k equations formed by equations 2 through k + 1 in (1. 5). We obtain an echelon form system of k equations in this way.

A„ £ R}. Thus Span(S) is the subspace ^ ( R ) C C(R). 9. 3. LINEAR COMBINATIONS 23 The fact that the span of a set of vectors is a subspace of the vector space from which the vectors are chosen is true in general. 4) T heorem . Let V be a vector space and let S be any subset of V. Then Span(S) is a subspace of V. 8) once again. SpanfS) is non­ empty by definition. Furthermore, let x, y G Span(S), and let c G R. Then we can write x = «,x, + • • ■ + a„x„, with a, G R and x, E S. Similarly, we can write y = fr,x| + • • ■ + b,„x'„, with b, E R and x' E S.

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