By Isaiah Lankham, Bruno Nachtergaele, Anne Schilling

ISBN-10: 9814730351

ISBN-13: 9789814730358

This is often an introductory textbook designed for undergraduate arithmetic majors with an emphasis on abstraction and particularly, the concept that of proofs within the atmosphere of linear algebra. often the sort of scholar may have taken calculus, notwithstanding the single prerequisite is appropriate mathematical grounding. the aim of this booklet is to bridge the distance among the extra conceptual and computational orientated undergraduate periods to the extra summary orientated periods. The booklet starts with structures of linear equations and complicated numbers, then relates those to the summary concept of linear maps on finite-dimensional vector areas, and covers diagonalization, eigenspaces, determinants, and the Spectral Theorem. every one bankruptcy concludes with either proof-writing and computational workouts.

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**Example text**

Now, assume z = 0, and set A = max{|a0 |, . . , |an−1 |}. We can obtain a lower bound for |f (z)| as follows: |f (z)| = |an | |z|n 1 + ≥ |an | |z| n an−1 1 a0 1 + ··· + an z an z n A 1− |an | ∞ k=1 1 1 A . = |an | |z|n 1 − |z|k |an | |z| − 1 For all z ∈ C such that |z| ≥ 2, we can further simplify this expression and obtain |f (z)| ≥ |an | |z|n 1 − 2A . |an ||z| It follows from this inequality that there is an R > 0 such that |f (z)| > |f (0)|, for all z ∈ C satisfying |z| > R. Let D ⊂ R2 be the disk of radius R centered at 0, and deﬁne a function g : D → R, by g(x, y) = |f (x + iy)|.

4. The vectors v1 = (1, 1, 1), v2 = (0, 1, −1), and v3 = (1, 2, 0) are linearly dependent. To see this, we need to consider the vector equation a1 v1 + a2 v2 + a3 v3 = a1 (1, 1, 1) + a2 (0, 1, −1) + a3 (1, 2, 0) = (a1 + a3 , a1 + a2 + 2a3 , a1 − a2 ) = (0, 0, 0). Solving for a1 , a2 , and a3 , we see, for example, that (a1 , a2 , a3 ) = (1, 1, −1) is a non-zero solution. Alternatively, we can reinterpret this vector equation as the homogeneous linear system ⎫ a1 + a3 = 0 ⎬ a1 + a2 + 2a3 = 0 . 3, we see that solving this linear system is equivalent to solving the following linear system: a1 + a3 = 0 a2 + a3 = 0 .

If U = U1 + U2 , then, for any u ∈ U , there exist u1 ∈ U1 and u2 ∈ U2 such that u = u1 + u2 . If it so happens that u can be uniquely written as u1 + u2 , then U is called the direct sum of U1 and U2 . page 34 November 2, 2015 14:50 ws-book961x669 Linear Algebra: As an Introduction to Abstract Mathematics Vector Spaces 9808-main 35 u+v ∈ / U ∪U u v U U Fig. 3. Suppose every u ∈ U can be uniquely written as u = u1 + u2 for u1 ∈ U1 and u2 ∈ U2 . Then we use U = U1 ⊕ U2 to denote the direct sum of U1 and U2 .

### Linear Algebra as an Introduction to Abstract Mathematics by Isaiah Lankham, Bruno Nachtergaele, Anne Schilling

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