Engineering Analysis

Engineering Analysis Chapter 5Orthogonality Basil Hamed

Orthogonality Definition. We say that 2 vectors are orthogonal if they are perpendicular to each other. i.e. the dot product of the two vectors is zero. ... A set of vectors S is orthonormal if every vector in S has magnitude 1 and the set of vectors are mutually orthogonal If the dot product of two vectors is 0, they are orthogonal, in other words, they are perpendicular. The dot product between two vectors u⃗ ,v⃗ is given by: u⃗ ⋅v⃗ =|u⃗ ||v⃗ |cos(θ), so when u⃗ ⋅v⃗ =0 ⟹ cosθ =0 ⟹ θ =π/2(90∘). Basil Hamed

then Basil Hamed

Basil Hamed

Definition Basil Hamed

If x and y are nonzero vectors, then we can specify their directions by forming unit vectors If θ is the angle between x and y, then The cosine of the angle between the vectors x and y is simply the scalar product of the corresponding direction vectors u and v. Basil Hamed

Basil Hamed

EXAMPLE 3 Let x and y be the vectors in Example 2. The directions of these vectors are given by the unit vectors The cosine of the angle θ between the two vectors is Basil Hamed

Basil Hamed

Scalar and Vector Projections Basil Hamed

Basil Hamed

EXAMPLE 5 The point Q in Figure 5.1.3 is the point on the line y = 1/3 x that is closest to the point (1, 4). Determine the coordinates of Q. Solution Thus, Q = (2.1, 0.7) is the closest point. Basil Hamed

Basil Hamed

The span of two linearly independent vectors x and y in R3 corresponds to a plane through the origin in 3-space. To determine the equation of the plane we must find a vector normal to the plane. In Section 3 of Chapter 2, it was shown that the cross product of the two vectors is orthogonal to each vector. If we take N = x × y as our normal vector, then the equation of the plane is given by Basil Hamed

EXAMPLE 7 Find the equation of the plane that passes through the points Solution Let The normal vector N must be orthogonal to both x and y. If we set We can then use any one of the points to determine the equation of the plane. Using the point P1, we see that the equation of the plane is Basil Hamed

Basil Hamed

If either x or y is the zero vector then x×y = 0 and hence the norm of x×y will be 0. Basil Hamed

5.2 Orthogonal Subspaces Definition Thus, Basil Hamed

5.2 Orthogonal Subspaces Basil Hamed

5.2 Orthogonal Subspaces Definition Basil Hamed

5.2 Orthogonal Subspaces Example Solution According to the proposition, we need to compute the null space of the matrix The free variable is x 3 , so the parametric form of the solution set is x 1 = x 3 / 17, x 2 = − 5 x 3 / 17, and the parametric vector form is Basil Hamed

5.2 Orthogonal Subspaces Scaling by a factor of 17, we see that We can check our work: Basil Hamed

5.2 Orthogonal Subspaces Example Find all vectors orthogonal to Solution According to the proposition, we need to compute the null space of the matrix This matrix is in reduced-row echelon form. The parametric form for the solution set is x 1 = − x 2 + x 3 , so the parametric vector form of the general solution is Therefore, the answer is the plane Basil Hamed

5.2 Orthogonal Subspaces Theorem 5.2.1 Fundamental Subspaces Theorem Let us denote the range of A by R(A). Thus Basil Hamed

5.2 Orthogonal Subspaces EXAMPLE 3 Let The column space of A consists of all vectors of the form Note that if x is any vector in R2 and b = Ax, then Basil Hamed

5.2 Orthogonal Subspaces Theorem 5.2.2 Definition Basil Hamed

5.2 Orthogonal Subspaces Theorem 5.2.4 EXAMPLE 4 Let Solution We can find bases for N(A) and R(AT ) by transforming A into reduced row echelon form: Basil Hamed

5.2 Orthogonal Subspaces Thus Basil Hamed

5.3 Least Squares Problems Basil Hamed

5.3 Least Squares Problems A standard technique in mathematical and statistical modeling is to find a least squares fit to a set of data points in the plane. The least squares curve is usually the graph of a standard type of function, such as a linear function, a polynomial, or a trigonometric polynomial. Since the data may include errors in measurement or experiment-related inaccuracies, we do not require the curve to pass through all the data points. Instead, we require the curve to provide an optimal approximation in the sense that the sum of squares of errors between the y values of the data points and the corresponding y values of the approximating curve are minimized. Basil Hamed

5.3 Least Squares Problems Least Squares Solutions of Overdetermined Systems A least squares problem can generally be formulated as an overdetermined linear system of equations. Recall that an overdetermined system is one involving more equations than unknowns. Such systems are usually inconsistent. Thus, given an m × n system Ax = b with m > n, we cannot expect in general to find a vector x ∈ Rn for which Ax equals b. Theorem 5.3.2 If A is an m × n matrix of rank n, the normal equations have a unique solution Basil Hamed

5.3 Least Squares Problems APPLICATION 2 Spring Constants Hooke’s law states that the force applied to a spring is proportional to the distance that the spring is stretched. Thus, if F is the force applied and x is the distance that the spring has been stretched, then F = kx. The proportionality constant k is called the spring constant. Some physics students want to determine the spring constant for a given spring. They apply forces of 3, 5, and 8 pounds, which have the effect of stretching the spring 4, 7, and 11 inches, respectively. Using Hooke’s law, they derive the following system of equations: Basil Hamed

5.3 Least Squares Problems The system is clearly inconsistent, since each equation yields a different value of k. Rather than use any one of these values, the students decide to compute the least squares solution of the system. Basil Hamed

5.3 Least Squares Problems EXAMPLE 1 Find the least squares solution of the system Solution The normal equations for this system are This simplifies to the 2 × 2 system The solution of the 2 × 2 system is Basil Hamed

5.3 Least Squares Problems Given a table of data we wish to find a linear function that best fits the data in the least squares sense. If we require that we get a system of m equations in two unknowns. The linear function whose coefficients are the least squares solution of (4) is said to be the best least squares fit to the data by a linear function. Basil Hamed

Engineering Analysis