= Let Q {q} be an orthonormal basis for a subspace V of R. a) Prove...
Fantastic news! We've Found the answer you've been seeking!
Question:
Transcribed Image Text:
= Let Q {q} be an orthonormal basis for a subspace V of R". a) Prove that if vЄV and v-q=0 for all i then v = 0. [2] 1 b) Prove that if v € V then v = (v qq (Hint. Take the difference and evaluate the inner products of it with the qj.) [3] c) Let V be a plane through the origin in R3 with orthonormal basis (q1,q2}. Recall that the orthogonal projection to the plane is defined by P(v) = v(v n)n where n is a unit normal to the plane. Show that P(v) = (v qi)q+(v q2)92 for all vЄ R³. [3] d) Let P be orthogonal projection to a plane V through the origin. in R³. Find the eigenvalues of P and show that there exists an orthonormal basis Q for R³ consisting of eigenvectors for P and find the matrix representation [P]o of P with respect to this basis. [4] e) Let T: R³ → R³ be the linear transformation of reflection through a plane V through the origin in R³. Find the eigenval- ues of T and an orthonormal basis for R³ consisting of eigen- vectors of T, and find the matrix representation [T]B of T with respect to this basis. [3] = Let Q {q} be an orthonormal basis for a subspace V of R". a) Prove that if vЄV and v-q=0 for all i then v = 0. [2] 1 b) Prove that if v € V then v = (v qq (Hint. Take the difference and evaluate the inner products of it with the qj.) [3] c) Let V be a plane through the origin in R3 with orthonormal basis (q1,q2}. Recall that the orthogonal projection to the plane is defined by P(v) = v(v n)n where n is a unit normal to the plane. Show that P(v) = (v qi)q+(v q2)92 for all vЄ R³. [3] d) Let P be orthogonal projection to a plane V through the origin. in R³. Find the eigenvalues of P and show that there exists an orthonormal basis Q for R³ consisting of eigenvectors for P and find the matrix representation [P]o of P with respect to this basis. [4] e) Let T: R³ → R³ be the linear transformation of reflection through a plane V through the origin in R³. Find the eigenval- ues of T and an orthonormal basis for R³ consisting of eigen- vectors of T, and find the matrix representation [T]B of T with respect to this basis. [3]
Expert Answer:
Related Book For
Posted Date:
Students also viewed these mathematics questions
-
Let u1, ...,uk be an orthonormal basis for the subspace W Rm. Let A = (u1, u2 ... uk) be the m à k matrix whose columns are the orthonormal basis vectors, and define P = AAT to be the...
-
Let B = {e1, e2,..., en} be an orthonormal basis of an inner product space V. Given T: V V, define T': V V by (a) Show that (aT)' = aT'. (b) Show that (5 + T)' = S' + T' (c) Show that MB(T') is the...
-
Let {x1, x2,..., xk, xk+1,..., xn} be an orthonormal basis for an inner product space V. Let S1 be the subspace of V spanned by x1,..., xk, and let S2 be the subspace spanned by xk+1, xk+2, ..., xn....
-
Which one the below does not define "Work role boundaries" of a care worker limits that allow a patient and staff to connect safely in a therapeutic relationship based on patients' needs rules of...
-
An analysis of the accounts receivable of Matsushita Company as of December 31, 20--, reveals the following: REQUIRED 1. Prepare an aging schedule as of December 31, 20--, by adding the following...
-
Figure P2.55 shows the motion diagram, made at two frames of film per second, of a ball rolling along a track. The track has a \(3.0-\mathrm{m}\)-long sticky section. a. Use the scale to determine...
-
The pie chart at the left shows the responses of a sample of 352 small-business owners who were asked whether their business has a website. Find the mean, the median, and the mode of the data, if...
-
Kenny Co. began operations on January 1, 2010, by issuing at $15 per share one half of the 950,000 shares of $1 par value common stock that had been authorized for sale. In addition, Kenny has...
-
The bill and memo: Read A.3252 of 2023, The Billionaire Mark-to-Market Tax Act, and the accompanying bill memo. Bill memos are not always faithful to the bill so be sure to read the bill carefully....
-
Captain America Corporation is a small private corporation that sells desktop printers to local businesses and schools. On May 1, 2021, the following were the account balances of Captain America...
-
Jessica has an insurance policy that will pay her an annual annuity of $580 at the end of each year for a total of 7 consecutive years. However, the terms of the policy are such that the annuity's...
-
We are going to play a game. First, follow the prompt and state your answer. Then use ChatGPT to provide you with a response to the prompt. Put both responses (yours and ChatGPT) in no particular...
-
A school-based health clinic (SBHC), located in an isolated region of the county has been serving high school students from the school and surrounding community for more than 15 years through...
-
Lorenzo shares the following information with you as you ponder different scenarios to help your client. Selling price $ 7 . 5 0 Cost for paper, per unit $ 0 . 7 0 Cost for printing, per unit $ 1 . 1...
-
Antonello Clothiers began operations on June 1 , 2 0 X 1 . During June, Antonello Clothiers reported the following: Purchases of merchandise inventory, at cost $ 1 1 0 , 0 0 0 Purchases returns, at...
-
5. A model airplane with a mass of 0.311 kg pulls out of a dive. The bottom of the dive is a circular arc with a radius of 24.6 m. At the bottom of the arc, the plane's speed is a constant 22.7 m/s....
-
The records of GHI Corp. reflected the following for the month of February: (15 points) Transaction Beginning inventory Number of Units Unit Cost 600 $3 Purchase No.1 $4 Sale No. 11 Purchase No. 2...
-
Dawson Companys balance sheet information at the end of 2019 and 2020 is as follows: Additional information: The company did not issue any common stock during 2020. Required : Next Level Fill in the...
-
Let A be the reduced incidence matrix for a structure and C the diagonal bar stiffness matrix. Suppose f is a set of external forces that maintains equilibrium of the structure. (a) Prove that f =...
-
(a) Prove that every real 3 3 matrix has at least one real eigenvalue. (b) Find a real 4 4 matrix with no real eigenvalues. (c) Can you find a real 5 5 matrix with no real eigenvalues?
-
A bank offers 5% interest compounded yearly. Suppose you deposit $120 in the account each year. Set up an affine iterative equation (10.5) to represent your bank balance. How much money do you have...
-
We are interested to produce $P$ in the reaction $A ightarrow P$ using a continuous reactor at $v=240$ liters/ hr with $C_{A_{0}}=3$ moles/liter. However, it is noticed that there is a second...
-
For installation and operation of a pipeline for an incompressible fluid, the total cost (in dollars per year) can be represented as follows: \[C=C_{1} D^{1.5} L+C_{2} m \Delta p / ho\] where...
-
An oil refinery has three process plants, and four grades of motor oil have been produced from these plants. The refinery is liable to meet the demand of customers. The refinery incurs a penalty for...
Study smarter with the SolutionInn App