By David B. Damiano

The authors outline basic vector areas and linear mappings on the outset and base all next advancements on those techniques. This process presents a ready-made context, motivation, and geometric interpretation for every new computational process. Proofs and summary problem-solving are brought from the beginning, providing scholars a right away chance to perform utilising what they have discovered. every one bankruptcy includes an creation, precis, and supplementary workouts. The textual content concludes with a couple of important appendixes and recommendations to chose exercises.

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**Example text**

Since this is true for all v E Span (5, U S2), we have Span(Si U S2) C W, + W2. Combining the two parts of the proof, we have the claim. ■ We now show that, in general, the sum of two subspaces is also a subspace. 9) Theorem . Let W, and W2 be subspaces of a vector space V. Then Wi + W2 is also a subspace of V. Proof: It is clear that W, + W2 is nonempty, since Wt and W2 are nonempty. Let x. y be any two vectors in W, + W2 and let c G R. Since x and y G W, + WL, we can write x = X| + x2. y = y, + y2, where x,.

DmI), ■ • • i (d 1«, . . ,, . . „ must be satisfied if the vector x E Span(S). ) Conversely if all these equations are satisfied, then there will be at least one solution of the system, so x will be in Span(S). The set of all equations of this form obtained from the echelon form system w ill be a set of defining equations for the subspace Span(S). 14) Exam ple. In R4 consider S = {(1,3, 1, 1), (1, —1 ,6 , 2)}. Then we have x = (b t, b2, b3, b4) E Span(S) if and only if the system jc[ + x2 = b\ 3x, — x2 = b2 x, + 6„v2 = b3 *i + 2x2 — b4 has solutions.

A,„, . . , and b= (b i , . . , b,„) £ R"', then b £ Span(S) if and only if there are scalars x t, . , x„ such that 115. i, . , bm) X\(ci\i, . , am\) + 33 + xn(a\fn . , amn) — (« nT| + • • • + ai„xn, . v„ = b, fl21-Yl + ' • ' + C l2 „ X n = b2 am\X\ + • • • + ci„mx„ - b,„ We ask, are there any solutions to the system? „,), what conditions must the fr, satisfy for solutions to exist ? Note that except for the possible nonzero right-hand sides b,. the form of these equations is the same as that of the ones previously considered.