# Find a Value of a Linear Transformation From $\R^2$ to $\R^3$

## Problem 142

Let $T:\R^2 \to \R^3$ be a linear transformation such that $T(\mathbf{e}_1)=\mathbf{u}_1$ and $T(\mathbf{e}_2)=\mathbf{u}_2$, where $\mathbf{e}_1=\begin{bmatrix} 1 \\ 0 \end{bmatrix}, \mathbf{e}_2=\begin{bmatrix} 0 \\ 1 \end{bmatrix}$ are unit vectors of $\R^2$ and
$\mathbf{u}_1= \begin{bmatrix} -1 \\ 0 \\ 1 \end{bmatrix}, \quad \mathbf{u}_2=\begin{bmatrix} 2 \\ 1 \\ 0 \end{bmatrix}.$ Then find $T\left(\begin{bmatrix} 3 \\ -2 \end{bmatrix}\right)$.

Contents

## Hint.

A linear transformation from a vector space $V$ to a vector space $W$ is a map $f:V \to W$ satisfying the following linearity properties:

1. $f(u+v)=f(u)+f(v)$ for any vectors $u, v \in V$, and
2. $f(rv)=rf(v)$ for any vector $v \in V$ and any scalar $r$.

Note that the set $\{\mathbf{e}_1, \mathbf{e}_2\}$ is a basis for the vector space $\R^2$.
Thus the vector $\begin{bmatrix} 3 \\ -2 \end{bmatrix}$ can be written as a linear combination of the basis vectors $\mathbf{e}_1, \mathbf{e}_2$.

## Solution.

We first express the vector $\begin{bmatrix} 3 \\ -2 \end{bmatrix}$ as a linear combination of $\mathbf{e}_1$ and $\mathbf{e}_2$:
$\begin{bmatrix} 3 \\ -2 \end{bmatrix}=3\mathbf{e}_1-2\mathbf{e}_2.$ Then we have
\begin{align*}
T\left(\begin{bmatrix}
3 \\
-2
\end{bmatrix}\right)
&=T(3\mathbf{e}_1-2\mathbf{e}_2)\\
&=3T(\mathbf{e}_1)-2T(\mathbf{e}_2) \text{ by the linearity of } T\\
&=3\mathbf{u}_1-2\mathbf{u}_2\\
&=3\begin{bmatrix}
-1 \\
0 \\
1
\end{bmatrix}-2\begin{bmatrix}
2 \\
1 \\
0
\end{bmatrix}\\
&=\begin{bmatrix}
-7 \\
-2 \\
3
\end{bmatrix}.
\end{align*}

Thus, we found
$T\left(\begin{bmatrix} 3 \\ -2 \end{bmatrix}\right) =\begin{bmatrix} -7 \\ -2 \\ 3 \end{bmatrix}.$

• Give the Formula for a Linear Transformation from $\R^3$ to $\R^2$ Let $T: \R^3 \to \R^2$ be a linear transformation such that $T(\mathbf{e}_1)=\begin{bmatrix} 1 \\ 4 \end{bmatrix}, T(\mathbf{e}_2)=\begin{bmatrix} 2 \\ 5 \end{bmatrix}, T(\mathbf{e}_3)=\begin{bmatrix} 3 \\ 6 […] • Vector Space of Polynomials and Coordinate Vectors Let P_2 be the vector space of all polynomials of degree two or less. Consider the subset in P_2 \[Q=\{ p_1(x), p_2(x), p_3(x), p_4(x)\},$ where \begin{align*} &p_1(x)=x^2+2x+1, &p_2(x)=2x^2+3x+1, \\ &p_3(x)=2x^2, &p_4(x)=2x^2+x+1. \end{align*} (a) Use the basis […]
• Linear Transformation to 1-Dimensional Vector Space and Its Kernel Let $n$ be a positive integer. Let $T:\R^n \to \R$ be a non-zero linear transformation. Prove the followings. (a) The nullity of $T$ is $n-1$. That is, the dimension of the nullspace of $T$ is $n-1$. (b) Let $B=\{\mathbf{v}_1, \cdots, \mathbf{v}_{n-1}\}$ be a basis of the […]
• Any Vector is a Linear Combination of Basis Vectors Uniquely Let $B=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ be a basis for a vector space $V$ over a scalar field $K$. Then show that any vector $\mathbf{v}\in V$ can be written uniquely as $\mathbf{v}=c_1\mathbf{v}_1+c_2\mathbf{v}_2+c_3\mathbf{v}_3,$ where $c_1, c_2, c_3$ are […]
• Determine linear transformation using matrix representation Let $T$ be the linear transformation from the $3$-dimensional vector space $\R^3$ to $\R^3$ itself satisfying the following relations. \begin{align*} T\left(\, \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} \,\right) =\begin{bmatrix} 1 \\ 0 \\ 1 […]
• Isomorphism of the Endomorphism and the Tensor Product of a Vector Space Let $V$ be a finite dimensional vector space over a field $K$ and let $\End (V)$ be the vector space of linear transformations from $V$ to $V$. Let $\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_n$ be a basis for $V$. Show that the map $\phi:\End (V) \to V^{\oplus n}$ defined by […]