Introduction to Linear Algebra 4th Edition solutions

Introduction to Linear Algebra 4
A typical point in R 3 is xi + y j + z k . The coordinate vectors i, j, and k are (1,0,0), (0, 1,0), (0,0, 1). The coordinates of the point are (x,y,z). This point in computer graphics is xi + y j + zk + origin. Its homogeneous coordinates are ( , , , ). Other coordinates for the same point are ( , ,

Introduction to Linear Algebra 4
A linear transformation T is determined when we know T(i), T(j), T(k). For an affine transformation we also need T ( ). The input point (x, y, z, 1) is transformed to xT(i) + yT(j) + zT(k) + _

Introduction to Linear Algebra 4
Multiply the 4 by 4 matrix T for translation along (1, 4, 3) and the matrix TI for translation along (0,2,5). The product TTl is translation along

Introduction to Linear Algebra 4
Write down the 4 by 4 matrix S that scales by a constant c. Multiply ST and also TS, where T is translation by (1,4,3). To blow up the picture around the center point (1,4,3), would you use vST or vTS?

Introduction to Linear Algebra 4
Write down the 4 by 4 matrix S that scales by a constant c. Multiply ST and also TS, where T is translation by (1,4,3). To blow up the picture around the center point (1,4,3), would you use vST or vTS?

Introduction to Linear Algebra 4
What 4 by 4 matrix would move a comer of a cube to the origin and then multiply all lengths by 2? The COlJler of the cube is originally at (1, 1,2).

Introduction to Linear Algebra 4
When the three matrices in equation 1 multiply the unit vector a, show that they give (cos e)a and (1- cos e)a and O. Addition gives aQ = a and the rotation axis is not moved

Introduction to Linear Algebra 4
If b is perpendicular to a, mUltiply by the three matrices in 1 to get (cos 8)b and 0 and a vector perpendicular to b. So Q b makes an angle e with b. This is rotation.

Introduction to Linear Algebra 4
What is the 3 by 3 projection matrix I - n n T onto the plane ~ x + ~ y + ~ z = O? In homogeneous coordinates add 0,0,0,1 as an extra row and column in P.

Introduction to Linear Algebra 4
With the same 4 by 4 matrix P, multiply T_PT+ to find the projection matrix onto the plane ~ x + ~ y + ~ z = 1. The translation T _ moves a point on that plane (choose one) to (0,0,0,1). The inverse matrix T+ moves it back.

Introduction to Linear Algebra 4
Project (3,3,3) onto those planes. Use P in Problem 9 and T_PT+ in Problem 10.

Introduction to Linear Algebra 4
If you project a square onto a plane, what shape do you get?

Introduction to Linear Algebra 4
If you project a cube onto a plane, what is the outline of the projection? Make the projection plane perpendicular to a diagonal of the cube.

Introduction to Linear Algebra 4
The 3 by 3 mirror matrix that reflects through the plane n T v = is M = I - 2n n T. Find the reflection of the point (3,3,3) in the plane ~x + ~ y + tz = 0.

Introduction to Linear Algebra 4
Find the reflection of (3,3,3) in the plane ~x + ~y + tz = 1. Take three steps T _ M T + using 4 by 4 matrices: translate by T _ so the plane goes through the origin, reflect the translated point (3, 3, 3, l)T_ in that plane, then translate back by T+.

Introduction to Linear Algebra 4
The vector between the origin (0,0,0, 1) and the point (x, y, z, 1) is the difference v = . In homogeneous coordinates, vectors end in . So we add a __ to a point, not a point to a point.

Introduction to Linear Algebra 4
If you multiply only the last coordinate of each point to get (x, y, z, c), you rescale the whole space by the number . This is because the point (x, y, z, c) is the same as ( , , ,1)