Answered: Problem 6.5. Bonus: It is known that… | bartleby

Linear Algebra: A Modern Introduction

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter4: Eigenvalues And Eigenvectors

Section4.2: Determinants

Problem 19EQ

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Problem 6.5. Bonus: It is known that the following matrix is an orthogonal projection on some two-
dimensional plane in R³. Compute x, y, z.
5
-2 x
Y
-2
-2
2

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Show that the three points (x1,y1)(x2,y2) and (x3,y3) in the a plane are collinear if and only if the matrix [x1y11x2y21x3y31] has rank less than 3.
4. (a) Given the points (1, 2), (2,6), and (4,1). Find the equation of the straight line that best fits these points. 1 1 1 2 (b) Find P, the projection matrix on C(A), where A = 1 4 (c) Verify that P2 = P.
Problem #10: Let W be the plane with the equation 7x-y + 5z = 0. Find the standard matrix P for the orthogonal projection onto W. Use the following formula P = = A(ATA) ¹AT, where the matrix A is constructed using any basis for Was its column vectors. Enter the first row of the matrix P (in order) into the answer box below, separated with commas. i.e., enter the values p11, P12, P13. Problem #10: Enter your answer symbolically, as in these examples
7. (a) Find the equation of the straight line y = ao + a₁t which best fits the following points: 12 13 4 t y 1 3 4 2 2 3 (b) Find the projection matrix onto C(A), the column space of A, where A = 1 1 1 2 1
12. The matrix that represents a reflection on the plane 2x – y + z = 0, is given (in the standard bases) by: -1 1 -1 1 (A) 3 -1 1 1 1. (B) It can't be computed with the information given
Q1. Consider the 3D shape with the following vertices: 4-38-60-60-98-88-8 A=5,B=5,C = 5,D = 5,E = 6,F= Rotate the 3D object by 30° (clockwise) at point D (fixed point) about the x-axis and then perform a uniform 1.5 scale. You should use rotation matrices.
2. Find the standard matrix of the orthogonal projection onto the image of 1 0 0 1 0 0 1 1 1 1 1 1 A = 1 1 1 1
6. (BH) For each of the following linear transformations, determine the kernel, the range, and write its standard matrix: (a) L: R3 + R3 projects every point x to the x2I3-plane (where r = 0). (b) L: R? + R² reflects a vector about the r2-axis and then rotates it by /4.
3. Describe all solutions of Ax = 0 in parametric vector form, where A is row equivalent to the given matrix. [1 -2 -9 5 012 -6
Problem 2.9. Let T: R³ R³ be a right-handed 90° rotation with axis the line x = y in the xy-plane (this line is spanned by the vector (1,1,0)). Find the matrix of T. Hint: First rotate about the z-axis by −45° to carry the line x = y to the x-axis, then rotate by 45° about the x-axis, and finally rotate the x-axis back to the line x = y.
5. Determine the matrix of the transformation that first reflects points through the horizontal x-axis and then reflects points through the line y = x.
Problem 25. Suppose yı (x), y2(x), and y3(x) are three different functions of x. The vector space they span could have dimension 1, 2, or 3. Give an example of y1, y2, y3 for each case.

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