Fundamentals of Matrix Algebra, 2011a

2.3 Visualizing Matrix Arithmec in 2D
Exercises 2.3
In Exercises 1 – 4, vectors ⃗x and ⃗y are given.
Sketch⃗x,⃗y,⃗x +⃗y, and⃗x −⃗y on the same Cartesian
axes.
[ ] [ ]
1 −2
1. ⃗x = ,⃗y =
1
2. ⃗x =
[ 3
1
]
,⃗y =
[ −1
3. ⃗x =
1
4. ⃗x =
]
,⃗y =
[ 2
0
]
,⃗y =
3
[ ] 1
−2
[ ] −2
2
[ ] 1
3
In Exercises 5 – 8, vectors ⃗x and ⃗y are drawn.
Sketch 2⃗x, −⃗y, ⃗x + ⃗y, and ⃗x − ⃗y on the same
Cartesian axes.
5.
6.
7.
8.
⃗y
1
.
⃗y
⃗y
1
1
.
1
.
⃗y
1
.
1
1
1
In Exercises 9 – 12, a vector ⃗x and a scalar
a are given. Using Definion 16, compute
⃗x
⃗x
⃗x
⃗x
the lengths of ⃗x and a⃗x, then compare these
lengths.
[ ] 2
9. ⃗x = , a = 3.
1
10. ⃗x =
[ −3
11. ⃗x =
5
12. ⃗x =
[ 4
7
]
, a = −2.
]
, a = −1.
[ 3
−9
]
, a = 1 3 .
13. Four pairs of vectors ⃗x and ⃗y are given
below. For each pair, compute ||⃗x||,
||⃗y||, and ||⃗x +⃗y||. Use this informaon
to answer: Is it always, somemes, or
never true that ||⃗x|| + ||⃗y|| = ||⃗x +⃗y||?
If it always or never true, explain why.
If it is somemes true, explain when it
is true.
[ ] [ ]
1 2
(a) ⃗x = ,⃗y =
1 3
[ ] [ ]
1 3
(b) ⃗x = ,⃗y =
−2 −6
[ ] [ ]
−1 2
(c) ⃗x = ,⃗y =
3 5
[ ] [ ]
2 −4
(d) ⃗x = ,⃗y =
1 −2
In Exercises 14 – 17, a matrix A is given.
Sketch ⃗x, ⃗y, A⃗x and A⃗y on the same Cartesian
axes, where
[ ] [ ]
1 −1
⃗x = and ⃗y = .
1 2
[ ] 1 −1
14. A =
2 3
[ ] 2 0
15. A =
−1 3
[ ] 1 1
16. A =
1 1
[ ] 1 2
17. A =
−1 −2
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