For Problems 1-7, determine whether the given set
S
of vectors is a basis for
ℝ
n
.
S
=
{
(
1
,
1
)
,
(
−
1
,
1
)
}
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
For each of the following lists of vectors in R3, determine whether the first vector can be expressed as a linear combination of the other two.
(a) (-2,0,3) ,(1,3,0),(2,4,-1)
(b) (1,2,-3) ,(-3,2,1) ,(2,-1,-1)
(c) (3,4,1) ,(1,-2,1), (-2,-1,1)
(d) (2,-1,0) , (1,2,-3), (1,-3,2)
(e) (5,1,-5) , (1,-2,-3), (-2,3,-4)
(f) (-2,2,2) ,(1,2,-1) ,(-3,-3,3)
1. Find all solutions:
2. In what ways can
88-8
and
be written
x+y ==0
a as a linear combination of [] [] [4], and ²
3. Show that the vector 2 is NOT contained in the span of the following three vectors
6
4. For which b are the vectors
-0.0
x+2y+z=5
2x+3y-z=3
independent?
A. Calculate the first two vectors generated by the Jacobi method.
[-4
-6
1
3
-1 x = {-6
x0) = 03x1
-1
5
7
Chapter 4 Solutions
Differential Equations and Linear Algebra (4th Edition)
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