Question 6.7.3: Let V be P2, with the inner product from Example 2, where t0...

Linear Algebra and Its Applications

Let V be $P _{2} \text {, with the inner product from Example 2, where } t_{0}=0$ , $t_{1}=\frac{1}{2}, \text { and } t_{2}=1 . \text { Let } p(t)=12 t^{2} \text { and } q(t)=2 t-1 . \text { Compute }\langle p, q\rangle \text { and }\langle q, q\rangle$ .

The "Step-by-Step Explanation" refers to a detailed and sequential breakdown of the solution or reasoning behind the answer. This comprehensive explanation walks through each step of the answer, offering you clarity and understanding.
Our explanations are based on the best information we have, but they may not always be right or fit every situation.

The blue check mark means that this solution has been answered and checked by an expert. This guarantees that the final answer is accurate.
Learn more on how we answer questions.

This Question has been Answered.

Already have an account?

Related Answered Questions

Question: 6.8.P.2

Find the first-order and third-order Fourier approximations to f (t ) = 3 – 2 sin t + 5 sin 2t – 6 cos 2t ...

Verified Answer:

The third-order Fourier approximation to f is the ...

Question: 6.8.P.1

Let q1(t) = 1, q2(t )= t , and q3(t ) = 3t ^2 – 4. Verify that {q1, q2, q3}is an orthogonal set in C[-2, 2] with the inner product of Example 7 in Section 6.7 (integration from -2 to 2). ...

Verified Answer:

Compute

\left\langle q_{1}, q_{2}\right\ran...

Question: 6.8.4

Find the nth-order Fourier approximation to the function f (t ) = t on the interval [0, 2π]. ...

Verified Answer:

Compute

\frac{a_{0}}{2}=\frac{1}{2} \cdot \...

Question: 6.8.3

Let C[0, 2π] have the inner product (6), and let m and n be unequal positive integers. Show that cos m t and cos nt are orthogonal. ...

Verified Answer:

\text { Use a trigonometric identity. When ...

Question: 6.8.2

The simplest and most common use of trend analysis occurs when the points t0,….,tn can be adjusted so that they are evenly spaced and sum to zero. Fit a quadratic trend function to the data (-2, 3), (-1, 5), (0, 5), (1, 4), and (2, 3). ...

Verified Answer:

The t -coordinates are suitably scaled to use the ...

Question: 6.8.1

Find the least-squares line y = β0 + β1x that best fits the data (-2, 3), (-1, 5), (0, 5), (1, 4), and (2,3). Suppose the errors in measuring the y-values of the last two data points are greater than for the other points. Weight these data half as much as the rest of the data. ...

Verified Answer:

As in Section 6.6, write X for the matrix A and β ...

Question: 6.7.P.1

Use the inner product axioms to verify the following statements. 1. 〈v, 0〉 = 〈0, v〉 = 0. 2. 〈u, v + w〉 = 〈u, v〉 + 〈u, w〉. ...

Verified Answer:

1.

\text { By Axiom } 1,\langle v , 0 \rangl...

Question: 6.6.5

In geography, local models of terrain are constructed from data (u1, v1, y1), …,(un, vn, yn), where uj , vj , and yj are latitude, longitude, and altitude, respectively. Describe the linear model based on (4) that gives a least-squares fit to such data. The solution is called the least-squares ...

Verified Answer:

We expect the data to satisfy the following equati...

Question: 6.7.8

Let V be the space C[0, 1] with the inner product of Example 7, and let W be the subspace spanned by the polynomials p1(t ) = 1, p2(t) = 2t – 1, and p3(t ) = 12t^2. Use the Gram–Schmidt process to find an orthogonal basis for W . ...

Verified Answer:

\text { Let } q_{1}=p_{1}, \text { and comp...

Question: 6.7.7

For f , g in C[a, b], set 〈 f,g 〉 = ∫a^b f(t)g(t ) dt Show that (5) defines an inner product on C[a, b]. ...

Verified Answer:

Inner product Axioms 1–3 follow from elementary pr...