Question

Problems 1. Determine whether the given transformation $T$ is linear. a) $T : M_{2,2} \to M_{2,2}$ defined by $$ T \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} a+b & 0 \\ 0 & c+d \end{pmatrix} $$ b) $T : P_2 \to P_2$ defined by $T(a + bx + cx^2) = (a - c) + b(x + 1) + b(x + 1)^2$ c) $T : M_{2,2} \to \mathbb{R}$ by $T(A) = \text{rank} (A)$ d) $T : \mathcal{F} \to \mathcal{F}$ defined by $T(f) = f(x^2)$ (where $\mathcal{F}$ is the vector space of all functions from $\mathbb{R}$ to $\mathbb{R}$)

          Problems
1. Determine whether the given transformation $T$ is linear.
a) $T : M_{2,2} \to M_{2,2}$ defined by
$$ T \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} a+b & 0 \\ 0 & c+d \end{pmatrix} $$
b) $T : P_2 \to P_2$ defined by $T(a + bx + cx^2) = (a - c) + b(x + 1) + b(x + 1)^2$
c) $T : M_{2,2} \to \mathbb{R}$ by $T(A) = \text{rank} (A)$
d) $T : \mathcal{F} \to \mathcal{F}$ defined by $T(f) = f(x^2)$ (where $\mathcal{F}$ is the vector space of all
functions from $\mathbb{R}$ to $\mathbb{R}$)

Problems
1. Determine whether the given transformation T is linear.
a) T : M2,2→ M2,2 defined by

T
< p m a t r i x >
=
< p m a t r i x >

b) T : P2 → P2 defined by T(a + bx + cx^2) = (a - c) + b(x + 1) + b(x + 1)^2
c) T : M2,2→ℝ by T(A) = rank (A)
d) T : ℱ→ℱ defined by T(f) = f(x^2) (where ℱ is the vector space of all
functions from ℝ to ℝ)

Added by David S.

Question

Please give Ace some feedback