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

Linear Transformation problems and solutions

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 $T\left(\begin{bmatrix}
3 \\
-2
\end{bmatrix}\right)$.

 
LoadingAdd to solve later

Sponsored Links


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
\[T\left(\begin{bmatrix}
3 \\
-2
\end{bmatrix}\right)
=\begin{bmatrix}
-7 \\
-2 \\
3
\end{bmatrix}.
\]


LoadingAdd to solve later

Sponsored Links

More from my site

  • Give the Formula for a Linear Transformation from $\R^3$ to $\R^2$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 VectorsVector 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 […]
  • Any Vector is a Linear Combination of Basis Vectors UniquelyAny 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 […]
  • Linear Transformation to 1-Dimensional Vector Space and Its KernelLinear 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 […]
  • Determine linear transformation using matrix representationDetermine 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 […]
  • Linear Transformation and a Basis of the Vector Space $\R^3$Linear Transformation and a Basis of the Vector Space $\R^3$ Let $T$ be a linear transformation from the vector space $\R^3$ to $\R^3$. Suppose that $k=3$ is the smallest positive integer such that $T^k=\mathbf{0}$ (the zero linear transformation) and suppose that we have $\mathbf{x}\in \R^3$ such that $T^2\mathbf{x}\neq \mathbf{0}$. Show […]
  • Isomorphism of the Endomorphism and the Tensor Product of a Vector SpaceIsomorphism 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 […]
  • Vector Space of Polynomials and a Basis of  Its SubspaceVector Space of Polynomials and a Basis of Its Subspace 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)=1, &p_2(x)=x^2+x+1, \\ &p_3(x)=2x^2, &p_4(x)=x^2-x+1. \end{align*} (a) Use the basis $B=\{1, x, […]

You may also like...

Leave a Reply

Your email address will not be published. Required fields are marked *

This site uses Akismet to reduce spam. Learn how your comment data is processed.

More in Linear Algebra
Linear Algebra Problems and Solutions
Linear Independent Vectors and the Vector Space Spanned By Them

Let $V$ be a vector space over a field $K$. Let $\mathbf{u}_1, \mathbf{u}_2, \dots, \mathbf{u}_n$ be linearly independent vectors in...

Close