Chapter 3.4

Question 1

If there is a line parallell to vector v that contains the point x0, what is the equation of the line?

Answer 1

x = x0 + tv

if x0 = 0 then the line passes through the origin and the equation is x = tv

Question 2

If there is a plane that is parallell to non collinear vectors v1 and v2 and cointains the point x0, what is the equation of the plane?

Answer 2

x = x0 +tv1 + tv2

if x+ = 0 then the plane passes through the origin and the equation is x = tv1 + tv2

Question 3

If x0 and v are vectors in R^(n) and v is nonzero, then what does the equation x = x0 + tv define?

Answer 3

A line through x0 that is parallell to v. In the special case of x0 = 0 the line is said to pass through the origin.

Question 4

What is said regarding the terminology of the equations

x = x0 + t1v1 + t2v2 + … + tnvn?

Answer 4

They are called the vector form of a subspace (line, plane or higher dimensional space).

If the corresponding equations on each side are equated, then the resulting equations are called parametric equations.

Question 5

Find vector and parametric equation of ax + by + cz +d = 0

Answer 5

1) Solve the equation for any one of the variables in terms of the other two and then use those two as variables. ie.

x = (-by-cz-d)/a
x = (-bt1-ct2-d)/a, y = t1, z = t2

2) To obtain a vector equation we rewrite these paramteric equations as:
(x, y, z) = ((-bt1-ct2-d)/a, t1, t2)
or equivalent
(x, y, z) = (-d/a, 0, 0) + t1(-b/a, 1, 0) + t2(-b/a, 0, 1)

Question 6

If x0 and x1 are vectors in R^(n) then what equations define the line segment from x0 to x1?

Answer 6

x = x0 + t(x1 - x0) (0 <=1)

Question 7

If A is an mxn matrix, what is the solution set of the homogeneous linear system Ax = 0 expressed through vectors?

Answer 7

The solution set consists of all the vectors in R^(n) that are orthogonal to every row vector of A

Question 8

If x0 and x1 are distinct points on a line in R^(n), what is the two-point vector equations for the line in R^(n)

Answer 8

x = x0 + t(x1 - x0)
or equivalent
x = (1 - t)x0 + tx1

Question 9

What are corresponding linear systems

Answer 9

Two linear systems with the same coefficientmatrices

Question 10

How can the general solution of Ax=b be obtained?

Answer 10

By adding a specific solution of Ax=b to the general solution Ax=0