By Pianigiani G.
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Extra resources for A Baire category approach in existence theory of differential equations
The process by which row operations are applied to a matrix to convert it to RREF is usually called Gauss– Jordan elimination. We will also say that we “row-reduced” a given matrix. While this process can be described in a somewhat cumbersome algorithm, it is best demonstrated with a few examples. By working through the details of the following problems (in particular by deciding which elementary row operations were performed at each stage), the reader will not only learn the basics of row reduction, but also will see and understand the key possibilities for the solution set of a system of linear equations.
Describe the set of all 8 −6 b2 vectors b for which b is a linear combination of v1 and v2 . 23. Let v1 = 32 Essentials of linear algebra 24. Let A be an m × n matrix, x and y ∈ Rn , and c ∈ R. Show that (a) A(x + y) = Ax + Ay (b) A(cx) = c(Ax) 25. Decide whether each of the following sentences is true or false. In every case, write one sentence to support your answer. (a) To compute the product Ax, the vector x must have the same number of entries as the number of rows in A. (b) A linear combination of three vectors in R3 produces another vector in R3 .
In particular, x is the vector in R2 whose components are x1 = 23 b1 − 13 b2 and x2 = − 13 b1 + 23 b2 . Thus the equation Ax = b is consistent for every b in R2 . 3, where we found that from a geometric perspective, every vector b ∈ R2 could be written as a linear combination of a1 and a2 . This example simply conﬁrms that ﬁnding, but now from an algebraic point of view. 3, we see that A has a pivot position in every row. In particular, there is no row in RREF(A) where we encounter all zeros, and thus it is impossible to ever encounter an equation of the form 0 = k, where k = 0.
A Baire category approach in existence theory of differential equations by Pianigiani G.