9.3. EXERCISES 197
13 z− 4
3 w
23 w− 5
3 z
zw
= z
13
− 53
10
+w
− 4
323
01
.
Example 9.2.15 The general solution of a linear system of equations is just the set of allsolutions. Find the general solution to the linear system,
1 2 3 02 1 1 24 5 7 2
xyzw
=
9725
given that(
1 1 2 1)T
=(
x y z w)T
is one solution.
Note the matrix on the left is the same as the matrix in Example 9.2.14. Therefore,from Theorem 9.2.13, you will obtain all solutions to the above linear system in the form
z
13
− 53
10
+w
− 4
323
01
+
1121
.
9.3 Exercises1. Study the definition of a linear transformation. State it from memory.
2. Show the map T : Rn 7→Rm defined by T (x) = Ax where A is an m×n matrix and xis an m×1 column vector is a linear transformation.
3. Find the matrix for the linear transformation which rotates every vector in R2 throughan angle of π/3.
4. Find the matrix for the linear transformation which rotates every vector in R2 throughan angle of π/4.
5. Find the matrix for the linear transformation which rotates every vector in R2 throughan angle of −π/3.
6. Find the matrix for the linear transformation which rotates every vector in R2 throughan angle of 2π/3.
7. Find the matrix for the linear transformation which rotates every vector in R2 throughan angle of π/12. Hint: Note that π/12 = π/3−π/4.
8. Find the matrix for the linear transformation which rotates every vector in R2 throughan angle of 2π/3 and then reflects across the x axis.