We can see this by seeing
\[ T(f+g)(x) = ( f(x) + g(x) )^2 = f^2(x) + 2 f(x) g(x) + g^2 (x) \]
while
\[ T(f)(x) + T(g)(x) = f^2(x) + g^2(x) . \]
Because $T(f+g) \neq T(f) + T(g)$, we see that $T$ is not a linear transformation.
For a specific example, consider $f(x) = g(x) = x$. Then
\[ T( f+g)(x) = (2x)^2 = 4x^2 , \]
while
\[ T(f)(x) + T(g)(x) = x^2 + x^2 = 2x^2 . \]
Clearly $T(f+g) \neq T(f) + T(g)$.
The Range and Nullspace of the Linear Transformation $T (f) (x) = x f(x)$
For an integer $n > 0$, let $\mathrm{P}_n$ be the vector space of polynomials of degree at most $n$. The set $B = \{ 1 , x , x^2 , \cdots , x^n \}$ is a basis of $\mathrm{P}_n$, called the standard basis.
Let $T : \mathrm{P}_n \rightarrow \mathrm{P}_{n+1}$ be the map defined by, […]
Find the Matrix Representation of $T(f)(x) = f(x^2)$ if it is a Linear Transformation
For an integer $n > 0$, let $\mathrm{P}_n$ denote the vector space of polynomials with real coefficients of degree $2$ or less. Define the map $T : \mathrm{P}_2 \rightarrow \mathrm{P}_4$ by
\[ T(f)(x) = f(x^2).\]
Determine if $T$ is a linear transformation.
If it is, find […]
Taking the Third Order Taylor Polynomial is a Linear Transformation
The space $C^{\infty} (\mathbb{R})$ is the vector space of real functions which are infinitely differentiable. Let $T : C^{\infty} (\mathbb{R}) \rightarrow \mathrm{P}_3$ be the map which takes $f \in C^{\infty}(\mathbb{R})$ to its third order Taylor polynomial, specifically defined […]
Find the Nullspace and Range of the Linear Transformation $T(f)(x) = f(x)-f(0)$
Let $C([-1, 1])$ denote the vector space of real-valued functions on the interval $[-1, 1]$. Define the vector subspace
\[W = \{ f \in C([-1, 1]) \mid f(0) = 0 \}.\]
Define the map $T : C([-1, 1]) \rightarrow W$ by $T(f)(x) = f(x) - f(0)$. Determine if $T$ is a linear map. If […]
Differentiation is a Linear Transformation
Let $P_3$ be the vector space of polynomials of degree $3$ or less with real coefficients.
(a) Prove that the differentiation is a linear transformation. That is, prove that the map $T:P_3 \to P_3$ defined by
\[T\left(\, f(x) \,\right)=\frac{d}{dx} f(x)\]
for any $f(x)\in […]
Are Linear Transformations of Derivatives and Integrations Linearly Independent?
Let $W=C^{\infty}(\R)$ be the vector space of all $C^{\infty}$ real-valued functions (smooth function, differentiable for all degrees of differentiation).
Let $V$ be the vector space of all linear transformations from $W$ to $W$.
The addition and the scalar multiplication of $V$ […]