6 6.1 © 2016 Pearson Education, Ltd. Orthogonality and Least Squares INNER PRODUCT, LENGTH, AND ORTHOGONALITY

 In order to find an approximate solution to an inconsistent system of equations that has no actual solution, a well-defined notion of nearness is needed  Distance, orthogonality (Sec. 6.1)  To identify the point within a subspace W that is nearest to a point lying outside of W (Sec )  Matrix factorization widely used in numerical linear algebra (Sec. 6.4)  Approximate solutions (Sec.6.5) Slide © 2012 Pearson Education, Ltd.

Slide © 2016 Pearson Education, Ltd. INNER PRODUCT

Slide INNER PRODUCT  If and, then the inner product of u and v is. © 2016 Pearson Education, Ltd.

Slide INNER PRODUCT © 2016 Pearson Education, Ltd.

Slide THE LENGTH OF A VECTOR © 2016 Pearson Education, Ltd.

Slide THE LENGTH OF A VECTOR  If we identify v with a geometric point in the plane, as usual, then coincides with the standard notion of the length of the line segment from the origin to v.  This follows from the Pythagorean Theorem applied to a triangle such as the one shown in the following figure.  For any scalar c, the length cv is times the length of v. That is, © 2016 Pearson Education, Ltd.

Slide THE LENGTH OF A VECTOR  A vector whose length is 1 is called a unit vector.  If we divide a nonzero vector v by its length—that is, multiply by —we obtain a unit vector u because the length of u is.  The process of creating u from v is sometimes called normalizing v, and we say that u is in the same direction as v. © 2016 Pearson Education, Ltd.

Slide THE LENGTH OF A VECTOR  Example 2: Let. Find a unit vector u in the same direction as v.  Solution: First, compute the length of v:  Then, multiply v by to obtain © 2016 Pearson Education, Ltd.

Slide © 2016 Pearson Education, Ltd.

Slide  Example 4: Compute the distance between the vectors and.  Solution: Calculate  The vectors u, v, and are shown in the figure on the next slide.  When the vector is added to v, the result is u. © 2016 Pearson Education, Ltd.

Slide  Notice that the parallelogram in the above figure shows that the distance from u to v is the same as the distance from to 0. © 2016 Pearson Education, Ltd.

Slide ORTHOGONAL VECTORS © 2016 Pearson Education, Ltd.

Slide ORTHOGONAL VECTORS Theorem 1(b) Theorem 1(a), (b) Theorem 1(a) © 2016 Pearson Education, Ltd.

Slide ORTHOGONAL VECTORS © 2016 Pearson Education, Ltd.

Slide THE PYTHOGOREAN THEOREM  Theorem 2: Two vectors u and v are orthogonal if and only if. © 2016 Pearson Education, Ltd.

Slide ORTHOGONAL COMPLEMENTS © 2016 Pearson Education, Ltd.

Slide ORTHOGONAL COMPLEMENTS © 2016 Pearson Education, Ltd.

Slide © 2012 Pearson Education, Ltd.

Slide ORTHOGONAL COMPLEMENTS © 2016 Pearson Education, Ltd.

Slide ORTHOGONAL COMPLEMENTS  Since this statement is true for any matrix, it is true for A T.  That is, the orthogonal complement of the row space of A T is the null space of A T.  This proves the second statement, because. © 2016 Pearson Education, Ltd.

Slide © 2016 Pearson Education, Ltd.

Slide  By the law of cosines, which can be rearranged to produce the equations on the next slide. © 2016 Pearson Education, Ltd.

Slide © 2016 Pearson Education, Ltd.