in exercises 3 and 4 display the following vectors using arrows on an

Question

In Exercises 3 and 4, display the following vectors using arrows
on an xy-graph: u, v, -v, -2v, u + v, u-v, and u-2v. Notice
that u - v is the vertex of a parallelogram whose other vertices are
u, 0, and -v.
3. u and v as in Exercise 1
4. u and v as in Exercise 2
In Exercises 5 and 6, write a system of equations that is equivalent
to the given vector equation.
5. X₁ -1
+ X₂
4
0
6. X₁
*[¯3 ] + x[$]+*[_!]-[8]
Use the accompanying figure to write each vector listed in Exer-
cises 7 and 8 as a linear combination of u and v. Is every vector
in R² a linear combination of u and v?
In Exercises 11 and 12, determine if b is a linear combination of
a₁, a₂, and a.
11. a₁ =
12. a₁ =
0
-------]
-6
8
=
----0--0-
=
13. A =
8
,b=
In Exercises 13 and 14, determine if b is a linear combination of
the vectors formed from the columns of the matrix A.
1-4 2
-40-0
3
5 ,b=
-2 8-4/n14. A =
1 -2 -6
0 3 7
-2 5
In Exercises 15 and 16, list five vectors in Span {V₁, V₂). For each
vector, show the weights on v₁ and v₂ used to generate the vector
and list the three entries of the vector. Do not make a sketch.
15. V₁ =
-0--0
-6
3
-19].M
2
16. V₁ =
HE]
b
V₂
-5
3/n19. Give a geometric description of Span (V₁, V₂} for the vectors
-[3]
V₁ =
2 and V₂ 1
20. Give a geometric description of Span (V₁, V₂) for the vectors
in Exercise 16.
21. Let u =
+-=[-²]
- [3].
and v=
Span (u, v) for all h and k.
Show that
is in
22. Construct a 3 x 3 matrix A, with nonzero entries, and a vector
b in R³ such that b is not in the set spanned by the columns
of A.
In Exercises 23 and 24, mark each statement True or False. Justify
each answer.
23. a. Another notation for the vector [] is [-4_3].
b. The points in the plane corresponding to [-]
[2] lie on a line through the origin.
and
c.
An example of a linear combination of vectors V₁ and V₂
is the vector V₁-
d. The solution set of the linear system whose augmented
matrix is [a₁ a2 a3 b] is the same as the solution
set of the equation x₁α₁ + X₂ả₂ + X3â3 = b.
e. The set Span (u, v) is always visualized as a plane
through the origin.
24. a.
Any list of five real numbers is a vector in R³.
b. The vector u results when a vector u - v is added to the
vector v.
c. The weights C₁,..., Cp in a linear combination
C₁V₁ ++ cpv, cannot all be zero.
d. When u and v are nonzero vectors, Span {(u, v) contains
the line through u and the origin.
e. Asking whether the linear system corresponding to
an augmented matrix [a a2 a b] has a solution
amounts to asking whether b is in Span (a₁, a₂, a3)./n2
10
3
26. Let A = -1
5, let b =
1
1
3
the set of all linear combinations of the columns of A.
a. Is b in W?
b. Show that the third column of A is in W.
0 6
8
-2
and let W be