Is the Map $T (f) (x) = f(x) – x – 1$ a Linear Transformation between Vector Spaces of Polynomials?

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• The Matrix Representation of the Linear Transformation $T (f) (x) = ( x^2 – 2) f(x)$ 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 $T : \mathrm{P}_3 \rightarrow \mathrm{P}_{5}$ be the map defined by, for $f \in […]
• 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 […]
• The Rank and Nullity of a Linear Transformation from Vector Spaces of Matrices to Polynomials Let $V$ be the vector space of $2 \times 2$ matrices with real entries, and $\mathrm{P}_3$ the vector space of real polynomials of degree 3 or less. Define the linear transformation $T : V \rightarrow \mathrm{P}_3$ by \[T \left( \begin{bmatrix} a & b \\ c & d \end{bmatrix} \right) = […]
• 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 […]
• Is the Map $T(f)(x) = f(0) + f(1) \cdot x + f(2) \cdot x^2 + f(3) \cdot x^3$ a Linear Transformation? Let $C ([0, 3] )$ be the vector space of real functions on the interval $[0, 3]$. Let $\mathrm{P}_3$ denote the set of real polynomials of degree $3$ or less. Define the map $T : C ([0, 3] ) \rightarrow \mathrm{P}_3 $ by \[T(f)(x) = f(0) + f(1) \cdot x + f(2) \cdot x^2 + f(3) […]
• Is the Following Function $T:\R^2 \to \R^3$ a Linear Transformation? Determine whether the function $T:\R^2 \to \R^3$ defined by \[T\left(\, \begin{bmatrix} x \\ y \end{bmatrix} \,\right) = \begin{bmatrix} x_+y \\ x+1 \\ 3y \end{bmatrix}\] is a linear transformation.   Solution. The […]
• A Linear Transformation Maps the Zero Vector to the Zero Vector Let $T : \mathbb{R}^n \to \mathbb{R}^m$ be a linear transformation. Let $\mathbf{0}_n$ and $\mathbf{0}_m$ be zero vectors of $\mathbb{R}^n$ and $\mathbb{R}^m$, respectively. Show that $T(\mathbf{0}_n)=\mathbf{0}_m$. (The Ohio State University Linear Algebra […]
• Prove that $\{ 1 , 1 + x , (1 + x)^2 \}$ is a Basis for the Vector Space of Polynomials of Degree $2$ or Less Let $\mathbf{P}_2$ be the vector space of polynomials of degree $2$ or less. (a) Prove that the set $\{ 1 , 1 + x , (1 + x)^2 \}$ is a basis for $\mathbf{P}_2$. (b) Write the polynomial $f(x) = 2 + 3x - x^2$ as a linear combination of the basis $\{ 1 , 1+x , (1+x)^2 […]