21. a. Find a homogeneous linear system of two equations in three unknowns whose solution space consists of those vectors in R that are orthogonal to the vectors a = (1,1,1) and b= (-2,3,0). b. What kind of geometric object is the solution space? c. Find a general solution of the system obtained in part (a). and confirm that Theorem 3.4.3 holds.
21. a. Find a homogeneous linear system of two equations in three unknowns whose solution space consists of those vectors in R that are orthogonal to the vectors a = (1,1,1) and b= (-2,3,0). b. What kind of geometric object is the solution space? c. Find a general solution of the system obtained in part (a). and confirm that Theorem 3.4.3 holds.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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PART (3.4) SECTION 21,22 WRITE ON PAPER PLEASE.
Transcribed Image Text:190 CHAPTER 3 Euclidean Vector Spaces
19. x₂ + 5x₂ + x3 + 2x₁ - x = 0
X₁-2x₂-x₂ + 3x + 2x, = 0
20. x₁ + 3x₂ - 4x, = 0
x + 2x + 3x, = 0
21. a. Find a homogeneous linear system of two equations in
three unknowns whose solution space consists of those
vectors in R that are orthogonal to the vectors a = (1,1,1)
and b = (-2,3,0).
b. What kind of geometric object is the solution space?
c. Find a general solution of the system obtained in part (a).
and confirm that Theorem 3.4.3 holds.
22. a. Find a homogeneous linear system of two equations in
three unknowns whose solution space consists of those
vectors in R³ that are orthogonal to a = (-3, 2, -1) and
b=(0, -2,-2).
b. What kind of geometric object is the solution space?
c. Find a general solution of the system obtained in part (a).
and confirm that Theorem 3.4.3 holds.
1 R be an
invertible matrix operator on R". Show that the image of a
line under multiplication by A is itself a line.
23. a. Let x = x + tv be a line in R" and let T₁: R"
b. Let TA: R² R² be multiplication by the matrix
-
A =
Find vector and parametric equations for the image under
multiplication by A of the line x = (1,3)+(2, -1).
24. Let TA: R³ → R³ be multiplication by the matrix
-4 3
A =
Find a vector equation for the image under multiplication by
A of the line segment
(x, y, z) = (1-1)(2.-3. 1)+(4.1.2) (0 <<1)
20220616_153136.jpg
True-False
TF. In parts
false, at
a. The
poi
line
b. The
any
Working
TI. Find t
and co
row v
Theor
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