C21 (pg498)

Determine if the linear transformation \(S:\quad { P }_{ 3 }\rightarrow { M }_{ 22 }\) is (a) injective, (b) surjective,(c) invertible.

\[S(a+bx+c{ x }^{ 2 }+d{ x }^{ 3 })=\left[ \begin{matrix} -a & +4b & +c & +2d \\ a & +5b & -2c & +2d \end{matrix}\quad \quad \begin{matrix} 4a & -b & +6c & -d \\ a & & +2c & +5d \end{matrix} \right]\]

# Creating matrix into S
(S <- matrix(c(-1,4,1,1,4,-1,5,0,1,6,-2,2,2,-1,2,5), 4,4))
##      [,1] [,2] [,3] [,4]
## [1,]   -1    4    1    2
## [2,]    4   -1    6   -1
## [3,]    1    5   -2    2
## [4,]    1    0    2    5
# RREF
(answer <- rref(S))
##      [,1] [,2] [,3] [,4]
## [1,]    1    0    0    0
## [2,]    0    1    0    0
## [3,]    0    0    1    0
## [4,]    0    0    0    1

Answer:

The identity matrix is nonsingular and have null space therefore it is injective. Uā€™s dimension is 4 and kernel is 0 therefore it is also surjective. Lastly, transformation is injective and surjective therefore it is also invertible as well as per ILTIS theorem.