# A Course in Linear Algebra by David B. Damiano

By David B. Damiano

Suitable for complex undergraduates and graduate scholars, this article deals an entire advent to the elemental innovations of linear algebra. fascinating and encouraging in its technique, it imparts an figuring out of the subject's logical constitution in addition to the ways that linear algebra presents ideas to difficulties in lots of branches of mathematics.
The authors outline common vector areas and linear mappings on the outset and base all next advancements on those techniques. This process offers a ready-made context, motivation, and geometric interpretation for every new computational procedure. Proofs and summary problem-solving are brought from the beginning, delivering scholars an instantaneous chance to perform employing what they have realized. every one bankruptcy includes an creation, precis, and supplementary routines. The textual content concludes with a couple of beneficial appendixes and options to chose exercises.

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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.