Linear algebra a modern introduction solution manual download

We have. But the final expression is nonnegative since it is a square. But the final expression is nonnegative since it is the sum of three squares. Thus these u. Let v1, v2,. Exploration: Vectors and Geometry » » » 1. Use the notation that the vector OX is written x. Draw AC. Also draw BD. Thus opposite sides of the quadrilateral P QRS are equal. Thus P QRS is a parallelogram.

Following the hint, we find m, the point that is two-thirds of the distance from A to P. From exercise 1, we have. Again from exercise 1, we have. Thus all the altitudes intersect at the orthocenter H. Thus RK 2. Let O, the center of the circle, be the origin.

We want to show that AC is orthogonal to BC. As in exercise 5, we first find m, the point that is halfway from P to R. Similarly, we find mj, the point that is halfway from Q to S. Any pair of points on A determine a direction vector, so we use P and Q.

We choose P to represent » the point on the line. We choose P to represent the » point on the line. We must find two direction vectors, u and v. Since u and v are not scalar multiples of each other, they will serve as direction vectors if they were parallel to each other, we would have not a plane but a line. This is true in general. So as t varies from 0 to 1, the line describes the line segment P Q.

Similarly, the vector whose head is 2 of the way from P to 3. Now, A1 and A2 are perpendicular if and only if their normal 1 vectors are perpendicular, i.

Suppose the line A has direction vector d, and the plane P has normal vector n. A is parallel to P. A is perpendicular to P. Suppose the plane P1 has normal vector n1 , and the plane P has normal vector n. P1 is parallel to P. A line is perpendicular to a plane if its direction vector d is the normal vector n of the plane. Note that a plane is parallel to a given plane if their normal vectors are equal. Using Figure 1. Now consider P4, the plane containing the face parallel to the face in the yz-plane but passing 1 through 1, 1, 1.

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Moreover, introducing vectors early allows students to see how systems of linear equations arise naturally from geometric problems. Matrices then arise equally naturally as coefficient matrices oflinear systems and as agents of change linear transformations. This sets the stage for eigenvectors and orthogonal projections, both of which are best understood geometrically. The dart that appears on the cover of this book symbolizes a vector and reflects my conviction that geometric understanding should precede computational techniques.

I have tried to limit the number of theorems in the text. For the most part, results labeled as theorems either will be used later in the text or summarize preceding work. Interesting results that are not central to the book have been included as exercises or explorations. For example, the cross product of vectors is discussed only in explorations in Chapters 1 and 4.

Unlike most linear algebra textbooks, this book has no chapter on determinants. The essential results are all in Section 4. The book is, however, comprehensive for an introductory text. I have not been stingy with the applications: There are many more in the book than can be covered in a single course.