Fundamentals of Matrix Algebra, 2011a

Chapter 2
Matrix Arithmec
.
Ḋefinion 16
Vector Length
Let
⃗x =
.
[
x1
]
.
x 2
The length of ⃗x, denoted ||⃗x||, is
√
||⃗x|| = x 2 1 + x2 2 .
. Example 38 Find the length of each of the vectors given below.
[ ] [ ] [ ] [ ]
1 2
.6 3
⃗x 1 = ⃗x
1 2 = ⃗x
−3 3 = ⃗x
.8 4 =
0
S
We apply Definion 16 to each vector.
.
√
||⃗x 1 || = 1 2 + 1 2 = √ 2.
√
||⃗x 2 || = 2 2 + (−3) 2 = √ 13.
√
||⃗x 3 || = .6 2 + .8 2 = √ .36 + .64 = 1.
√
||⃗x 4 || = 3 2 + 0 = 3.
Now that we know how to compute the length of a vector, let’s revisit a statement
we made as we explored Examples 34 and 35: “Mulplying a vector by a posive scalar
c stretches the vectors by a factor of c . . .” At that me, we did not know how to measure
the length of a vector, so our statement was unfounded. In the following example,
we will confirm the truth of our previous statement.
[ ]
. 2
Example 39 .Let ⃗x = . Compute ||⃗x||, ||3⃗x||, || − 2⃗x||, and ||c⃗x||, where c
−1
is a scalar.
S
We apply Definion 16 to each of the vectors.
||⃗x|| = √ 4 + 1 = √ 5.
Before compung the length of ||3⃗x||, we note that 3⃗x =
||3⃗x|| = √ 36 + 9 = √ 45 = 3 √ 5 = 3||⃗x||.
[
6
−3
]
.
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