宁波诺丁汉大学《线性代数》课件-CHAPTER 6.pptVIP

§6.3 Orthogonal Projections §6.4 The Gram – Schmidt Process What is Gram – Schmidt Process? Goal: Form an orthogonal basis for a subspace W. So {v1, v2} is an orthogonal basis for W. §6.2 Orthogonal Sets §6.2 Orthogonal Sets Definition： Proof： As in the preceding proof, the orthogonality of {u1, …, up} show that y?u1= (c1u1 + c2u2 +…+ cpup)?u1 = c1 (u1 ?u1) Since u1?u1 is not zero, the equation above can be solved for c1. To find cj for j=2, … ,p, compute y?uj and solve cj. Example：The set S={u1, u2, u3} in the above example is an orthogonal basis for R3. Express the vector as a linear combination of the vectors in S. Solution：Compute from Theorem 5, 2. An Orthogonal Projection Example： let and . Find the orthogonal projection of y onto u. then write y as the sum of two orthogonal vectors, one in Span {u} and one orthogonal to u. Solution: Compute The orthogonal projection of y onto u is And the component of y orthogonal to u is The orthogonal projection of y onto u is The decomposition of y is illustrated in the following Fig. Note: If the calculations above are correct, then will be an orthogonal set. As a check, compute： §6.2 Orthogonal Sets Example：Find the distance in Fig.3 from y to L. Solution：The distance from y to L is the length of the perpendicular line segment from y to the orthogonal projection . This length equals the length of . Thus the distance is §6.2 Orthogonal Sets 3. Orthonormal Sets (单位正交集) §6.2 Orthogonal Sets Example：Show that {v1, v2, v3} is an orthognomal basis of R3, where Solution：Compute Thus {v1, v2, v3} is an orthogonal set. Also Which shows that v1, v2, and v3 are unit vectors. Thus {v1, v2, v3} is an orthonormal set. Since the set is linearly independent, its three vectors from a basis for R3. §6.2 Orthogonal Sets §6.2 Orthogonal Sets Example ：let and Notice that U has orthonormal columns and Verify that Solution： §6.3 Orthog