Fundamentals of Matrix Algebra, 2011a

Before compung the length of || − 2⃗x||, we note that −2⃗x =
|| − 2⃗x|| = √ 16 + 4 = √ 20 = 2 √ 5 = 2||⃗x||.
[ ] 2c
Finally, to compute ||c⃗x||, we note that c⃗x = . Thus:
−c
||c⃗x|| = √ (2c) 2 + (−c) 2 = √ 4c 2 + c 2 = √ 5c 2 = |c| √ 5.
2.3 Visualizing Matrix Arithmec in 2D
[ ] −4
.
2
This last line is true because the square root of any number squared is the absolute
value of that number (for example, √ (−3) 2 = 3). .
The last computaon of our example is the most important one. It shows that,
in general, mulplying a vector ⃗x by a scalar c stretches ⃗x by a factor of |c| (and the
direcon will change if c is negave). This is important so we’ll make it a Theorem.
.
Ṫheorem 4
Vector Length and Scalar Mulplicaon
.
Let ⃗x be a vector and let c be a scalar. Then the length of c⃗x
is
||c⃗x|| = |c| · ||⃗x||.
Matrix − Vector Mulplicaon
The last arithmec operaon to consider visualizing is matrix mulplicaon. Specifically,
we want to visualize the result of mulplying a vector by a matrix. In order to
mulply a 2D vector by a matrix and get a 2D vector back, our matrix must be a square,
2 × 2 matrix. 11
We’ll start with an example. Given a matrix A and several vectors, we’ll graph the
vectors before and aer they’ve been mulplied by A and see what we learn.
. Example 40 .Let A be a matrix, and ⃗x,⃗y, and⃗z be vectors as given below.
A =
[ ] [ ]
1 4 1
, ⃗x = , ⃗y =
2 3 1
Graph ⃗x,⃗y and⃗z, as well as A⃗x, A⃗y and A⃗z.
[ ] [ ]
−1
3
, ⃗z =
1
−1
S
11 We can mulply a 3 × 2 matrix by a 2D vector and get a 3D vector back, and this gives very interesng
results. See secon 5.2.
75