Linear Transformation from $\R^n$ to $\R^m$

Linear Transformation from $\R^n$ to $\R^m$

Definition

  1. A function $T:\R^n \to \R^m$ is called a linear transformation if $T$ satisfies the following two linearity conditions: For any $\mathbf{x}, \mathbf{y}\in \R^n$ and $c\in \R$, we have
    1. $T(\mathbf{x}+\mathbf{y})=T(\mathbf{x})+T(\mathbf{y})$
    2. $T(c\mathbf{x})=cT(\mathbf{x})$
  2. The nullspace $\calN(T)$ of a linear transformation $T:\R^n\to \R^m$ is
    \[\calN(T)=\{\mathbf{x}\in \R^n \mid T(\mathbf{x})=\mathbf{0}_m\}.\]
  3. The nullity of $T$ is the dimension of $\calN(T)$.
  4. The range $\calR(T)$ of a linear transformation $T:\R^n\to \R^m$ is
    \[\calR(T)=\{\mathbf{y}\in \R^m \mid \mathbf{y}=T(\mathbf{x}) \text{ for some } \mathbf{x}\in \R^n\}.\]
  5. The rank of $T$ is the dimension of $\calR(T)$.
  6. The matrix representation of a linear transformation $T:\R^n \to \R^m$ is an $m\times n$ matrix $A$ such that $T(\mathbf{x})=A\mathbf{x}$ for all $\mathbf{x}\in \R^n$.
Summary

Let $T:\R^n \to \R^m$ be a linear transformation.

  1. $T(\mathbf{0}_n)=\mathbf{0}_m$, where $\mathbf{0}_n$ and $\mathbf{0}_m$ are the zero vectors in $\R^n$ and $R^m$, respectively.
  2. The matrix representation $A$ of a linear transformation $T:\R^n \to \R^m$ is given by $A=[T(\mathbf{e}_1), \dots, T(\mathbf{e}_n)]$, where $\mathbf{e}_1, \dots, \mathbf{e}_n$ are the standard basis for $\R^n$.
  3. If $A$ is the matrix representaiton of a linear transformation $T$, then
    1. $\calN(T)=\calN(A)$ and $\calR(T)=\calR(A)$.
    2. The nullity of $T$ is the same as the nullity of $A$.
    3. The rank of $T$ is the same as the rank of $A$.

=solution

Problems

  1. Let $T:\R^2 \to \R^3$ be a linear transformation such that
    \[T(\mathbf{e}_1)=\mathbf{u}_1 \text{ and } T(\mathbf{e}_2)=\mathbf{u}_2,\] where $\{\mathbf{e}_1, \mathbf{e}_2\}$ is the standard unit vectors of $\R^2$ and
    \[\mathbf{u}_1=\begin{bmatrix}
    5 \\
    1 \\
    2
    \end{bmatrix} \text{ and } \mathbf{u}_2=\begin{bmatrix}
    8 \\
    2 \\
    6
    \end{bmatrix}.\] Then find $T\left(\, \begin{bmatrix}
    3 \\
    -2
    \end{bmatrix} \,\right)$.

  2. 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 $T(\mathbf{0}_n)=\mathbf{0}_m$.

  3. 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.

  4. Let $T: \R^2 \to \R^2$ be a linear transformation.
    Let
    \[
    \mathbf{u}=\begin{bmatrix}
    1 \\
    2
    \end{bmatrix}, \mathbf{v}=\begin{bmatrix}
    3 \\
    5
    \end{bmatrix}\] be 2-dimensional vectors. Suppose that
    \begin{align*}
    T(\mathbf{u})&=T\left( \begin{bmatrix}
    1 \\
    2
    \end{bmatrix} \right)=\begin{bmatrix}
    -3 \\
    5
    \end{bmatrix},\\
    T(\mathbf{v})&=T\left(\begin{bmatrix}
    3 \\
    5
    \end{bmatrix}\right)=\begin{bmatrix}
    7 \\
    1
    \end{bmatrix}.
    \end{align*}
    Let $\mathbf{w}=\begin{bmatrix}
    x \\
    y
    \end{bmatrix}\in \R^2$. Find the formula for $T(\mathbf{w})$ in terms of $x$ and $y$.

  5. Let $\{\mathbf{v}_1, \mathbf{v}_2\}$ be a basis of the vector space $\R^2$, where $\mathbf{v}_1=\begin{bmatrix}
    1 \\
    1
    \end{bmatrix}$ and $\mathbf{v}_2=\begin{bmatrix}
    1 \\
    -1
    \end{bmatrix}$. The action of a linear transformation $T:\R^2\to \R^3$ on the basis $\{\mathbf{v}_1, \mathbf{v}_2\}$ is given by
    \begin{align*}
    T(\mathbf{v}_1)=\begin{bmatrix}
    2 \\
    4 \\
    6
    \end{bmatrix} \text{ and } T(\mathbf{v}_2)=\begin{bmatrix}
    0 \\
    8 \\
    10
    \end{bmatrix}.
    \end{align*}
    Find the formula of $T(\mathbf{x})$, where $\mathbf{x}=\begin{bmatrix}
    x \\
    y
    \end{bmatrix}\in \R^2$.

  6. Let $T: \R^3 \to \R^2$ be a linear transformation such that
    \[T(\mathbf{e}_1)=\begin{bmatrix}
    1 \\
    4
    \end{bmatrix}, T(\mathbf{e}_2)=\begin{bmatrix}
    2 \\
    5
    \end{bmatrix}, T(\mathbf{e}_3)=\begin{bmatrix}
    3 \\
    6
    \end{bmatrix},\] where
    \[\mathbf{e}_1=\begin{bmatrix}
    1 \\
    0 \\
    0
    \end{bmatrix}, \mathbf{e}_2=\begin{bmatrix}
    0 \\
    1 \\
    0
    \end{bmatrix}, \mathbf{e}_3=\begin{bmatrix}
    0 \\
    0 \\
    1
    \end{bmatrix}\] are the standard unit basis vectors of $\R^3$.
    For any vector $\mathbf{x}=\begin{bmatrix}
    x_1 \\
    x_2 \\
    x_3
    \end{bmatrix}\in \R^3$, find a formula for $T(\mathbf{x})$.

  7. Let $T:\R^2\to \R^2$ be a linear transformation such that it maps the vectors $\mathbf{v}_1, \mathbf{v}_2$ as indicated in the figure below.

    linear transformation from R^2 to R^2

    Find the matrix representation $A$ of the linear transformation $T$.

  8. Let $T$ be a linear transformation from $\R^3$ to $\R^2$ such that
    \[ T\left(\, \begin{bmatrix}
    0 \\
    1 \\
    0
    \end{bmatrix}\,\right) =\begin{bmatrix}
    1 \\
    2
    \end{bmatrix} \text{ and }T\left(\, \begin{bmatrix}
    0 \\
    1 \\
    1
    \end{bmatrix}\,\right)=\begin{bmatrix}
    0 \\
    1
    \end{bmatrix}. \] Then find $T\left(\, \begin{bmatrix}
    0 \\
    1 \\
    2
    \end{bmatrix} \,\right)$.
    (The Ohio State University)

  9. Let $T:\R^3 \to \R^2$ be a linear transformation such that
    \[ T(\mathbf{e}_1)=\begin{bmatrix}
    1 \\
    0
    \end{bmatrix}, T(\mathbf{e}_2)=\begin{bmatrix}
    0 \\
    1
    \end{bmatrix}, T(\mathbf{e}_3)=\begin{bmatrix}
    1 \\
    0
    \end{bmatrix},\] where $\mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3$ are the standard basis of $\R^3$.
    Then find the rank and the nullity of $T$.
    (The Ohio State University)

  10. Let $T:\R^2 \to \R^3$ be a linear transformation such that
    \[T\left(\, \begin{bmatrix}
    3 \\
    2
    \end{bmatrix} \,\right)
    =\begin{bmatrix}
    1 \\
    2 \\
    3
    \end{bmatrix} \text{ and }
    T\left(\, \begin{bmatrix}
    4\\
    3
    \end{bmatrix} \,\right)
    =\begin{bmatrix}
    0 \\
    -5 \\
    1
    \end{bmatrix}.\] (a) Find the matrix representation of $T$ (with respect to the standard basis for $\R^2$).
    (b) Determine the rank and nullity of $T$.
    (The Ohio State University)

  11. Define the map $T:\R^2 \to \R^3$ by $T \left ( \begin{bmatrix}
    x_1 \\
    x_2
    \end{bmatrix}\right )=\begin{bmatrix}
    x_1-x_2 \\
    x_1+x_2 \\
    x_2
    \end{bmatrix}$.
    (a) Show that $T$ is a linear transformation.
    (b) Find a matrix $A$ such that $T(\mathbf{x})=A\mathbf{x}$ for each $\mathbf{x} \in \R^2$.
    (c) Describe the null space (kernel) and the range of $T$ and give the rank and the nullity of $T$.

  12. Let $T:\R^4 \to \R^3$ be a linear transformation defined by
    \[ T\left (\, \begin{bmatrix}
    x_1 \\
    x_2 \\
    x_3 \\
    x_4
    \end{bmatrix} \,\right) = \begin{bmatrix}
    x_1+2x_2+3x_3-x_4 \\
    3x_1+5x_2+8x_3-2x_4 \\
    x_1+x_2+2x_3
    \end{bmatrix}.\] (a) Find a matrix $A$ such that $T(\mathbf{x})=A\mathbf{x}$.
    (b) Find a basis for the null space of $T$.
    (c) Find the rank of the linear transformation $T$.
    (The Ohio State University)

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

  14. Let $B=\{\mathbf{v}_1, \mathbf{v}_2 \}$ be a basis for the vector space $\R^2$, and let $T:\R^2 \to \R^2$ be a linear transformation such that
    \[T(\mathbf{v}_1)=\begin{bmatrix}
    1 \\
    -2
    \end{bmatrix} \text{ and } T(\mathbf{v}_2)=\begin{bmatrix}
    3 \\
    1
    \end{bmatrix}.\] If $\mathbf{e}_1=\mathbf{v}_1+2\mathbf{v}_2 \text{ and } \mathbf{e}_2=2\mathbf{v}_1-\mathbf{u}_2$, where $\mathbf{e}_1, \mathbf{e}_2$ are the standard unit vectors in $\R^2$, then find the matrix of $T$ with respect to the basis $\{\mathbf{e}_1, \mathbf{e}_2\}$.

  15. Let $T: \R^2 \to \R^2$ be a linear transformation such that
    \[T\left(\, \begin{bmatrix}
    1 \\
    1
    \end{bmatrix} \,\right)=\begin{bmatrix}
    4 \\
    1
    \end{bmatrix}, T\left(\, \begin{bmatrix}
    0 \\
    1
    \end{bmatrix} \,\right)=\begin{bmatrix}
    3 \\
    2
    \end{bmatrix}.\] Then find the matrix $A$ such that $T(\mathbf{x})=A\mathbf{x}$ for every $\mathbf{x}\in \R^2$, and find the rank and nullity of $T$.
    (The Ohio State University)

  16. If $L:\R^2 \to \R^3$ is a linear transformation such that
    \begin{align*}
    L\left( \begin{bmatrix}
    1 \\
    0
    \end{bmatrix}\right)
    =\begin{bmatrix}
    1 \\
    1 \\
    2
    \end{bmatrix}, \,\,\,\,
    L\left( \begin{bmatrix}
    1 \\
    1
    \end{bmatrix}\right)
    =\begin{bmatrix}
    2 \\
    3 \\
    2
    \end{bmatrix}.
    \end{align*}
    then
    (a) find $L\left( \begin{bmatrix}
    1 \\
    2
    \end{bmatrix}\right)$, and
    (b) find the formula for $L\left( \begin{bmatrix}
    x \\
    y
    \end{bmatrix}\right)$.
    (Purdue University)

  17. Suppose that $T: \R^2 \to \R^3$ is a linear transformation satisfying
    \[T\left(\, \begin{bmatrix}
    1 \\
    2
    \end{bmatrix}\,\right)=\begin{bmatrix}
    3 \\
    4 \\
    5
    \end{bmatrix} \text{ and } T\left(\, \begin{bmatrix}
    0 \\
    1
    \end{bmatrix} \,\right)=\begin{bmatrix}
    0 \\
    0 \\
    1
    \end{bmatrix}.\] Find a general formula for $T\left(\, \begin{bmatrix}
    x_1 \\
    x_2
    \end{bmatrix} \,\right)$.
    (The Ohio State University)

  18. 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
    \end{bmatrix}, \qquad T\left(\, \begin{bmatrix}
    2 \\
    3 \\
    5
    \end{bmatrix} \, \right) =
    \begin{bmatrix}
    0 \\
    2 \\
    -1
    \end{bmatrix}, \qquad
    T \left( \, \begin{bmatrix}
    0 \\
    1 \\
    2
    \end{bmatrix} \, \right)=
    \begin{bmatrix}
    1 \\
    0 \\
    0
    \end{bmatrix}.
    \end{align*}
    Then for any vector $\mathbf{x}=\begin{bmatrix}
    x \\
    y \\
    z
    \end{bmatrix}\in \R^3$, find the formula for $T(\mathbf{x})$.

  19. Let $T:\R^2 \to \R^3$ be a linear transformation given by
    \[T\left(\, \begin{bmatrix}
    x_1 \\
    x_2
    \end{bmatrix} \,\right)
    =
    \begin{bmatrix}
    x_1-x_2 \\
    x_2 \\
    x_1+ x_2
    \end{bmatrix}.\] Find an orthonormal basis of the range of $T$.
    (The Ohio State University)

  20. Let $T: \R^n \to \R^m$ be a linear transformation. Suppose that $S=\{\mathbf{x}_1, \mathbf{x}_2,\dots, \mathbf{x}_k\}$ is a subset of $\R^n$ such that $\{T(\mathbf{x}_1), T(\mathbf{x}_2), \dots, T(\mathbf{x}_k) \}$ is a linearly independent subset of $\R^m$. Prove that the set $S$ is linearly independent.

  21. 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 that the vectors $\mathbf{x}, T\mathbf{x}, T^2\mathbf{x}$ form a basis for $\R^3$.
    (The Ohio State University)

  22. 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 nullspace $\calN(T)$ of $T$. Let $\mathbf{w}$ be the $n$-dimensional vector that is not in $\calN(T)$. Then $B’=\{\mathbf{v}_1, \cdots, \mathbf{v}_{n-1}, \mathbf{w}\}$ is a basis of $\R^n$.
    (c) Each vector $\mathbf{u}\in \R^n$ can be expressed as
    \[\mathbf{u}=\mathbf{v}+\frac{T(\mathbf{u})}{T(\mathbf{w})}\mathbf{w}\] for some vector $\mathbf{v}\in \calN(T)$.

  23. Let $V$ be the subspace of $\R^4$ defined by the equation
    \[x_1-x_2+2x_3+6x_4=0.\] Find a linear transformation $T$ from $\R^3$ to $\R^4$ such that the null space $\calN(T)=\{\mathbf{0}\}$ and the range $\calR(T)=V$. Describe $T$ by its matrix $A$.

  24. Let $\mathbf{u}=\begin{bmatrix}
    1 \\
    1 \\
    0
    \end{bmatrix}$ and $T:\R^3 \to \R^3$ be the linear transformation
    \[T(\mathbf{x})=\proj_{\mathbf{u}}\mathbf{x}=\left(\, \frac{\mathbf{u}\cdot \mathbf{x}}{\mathbf{u}\cdot \mathbf{u}} \,\right)\mathbf{u}.\] (a) Calculate the null space $\calN(T)$, a basis for $\calN(T)$ and nullity of $T$.
    (b) Only by using part (a) and no other calculations, find $\det(A)$, where $A$ is the matrix representation of $T$ with respect to the standard basis of $\R^3$.
    (c) Calculate the range $\calR(T)$, a basis for $\calR(T)$ and the rank of $T$.
    (d) Calculate the matrix $A$ representing $T$ with respect to the standard basis for $\R^3$.
    (e) Let $B=\left\{\, \begin{bmatrix}
    1 \\
    0 \\
    0
    \end{bmatrix}, \begin{bmatrix}
    -1 \\
    1 \\
    0
    \end{bmatrix}, \begin{bmatrix}
    0 \\
    -1 \\
    1
    \end{bmatrix} \,\right\}$ be a basis for $\R^3$. Calculate the coordinates of $\begin{bmatrix}
    x \\
    y \\
    z
    \end{bmatrix}$ with respect to $B$.
    (The Ohio State University)

  25. We fix a nonzero vector $\mathbf{a}$ in $\R^3$ and define a map $T:\R^3\to \R^3$ by
    \[T(\mathbf{v})=\mathbf{a}\times \mathbf{v}\] for all $\mathbf{v}\in \R^3$. Here the right-hand side is the cross product of $\mathbf{a}$ and $\mathbf{v}$.
    (a) Prove that $T:\R^3\to \R^3$ is a linear transformation.
    (b) Determine the eigenvalues and eigenvectors of $T$.

  26. Determine all linear transformations of the $2$-dimensional $x$-$y$ plane $\R^2$ that take the line $y=x$ to the line $y=-x$.
  27. Let $F:\R^2\to \R^2$ be the function that maps each vector in $\R^2$ to its reflection with respect to $x$-axis. Determine the formula for the function $F$ and prove that $F$ is a linear transformation.
  28. Let $T:\R^2 \to \R^2$ be a linear transformation of the $2$-dimensional vector space $\R^2$ (the $x$-$y$-plane) to itself which is the reflection across a line $y=mx$ for some $m\in \R$. Then find the matrix representation of the linear transformation $T$ with respect to the standard basis $B=\{\mathbf{e}_1, \mathbf{e}_2\}$ of $\R^2$, where $\mathbf{e}_1=\begin{bmatrix}
    1 \\
    0
    \end{bmatrix}, \mathbf{e}_2=\begin{bmatrix}
    0 \\
    1
    \end{bmatrix}$.

  29. Let $T:\R^3 \to \R^3$ be a linear transformation and suppose that its matrix representation with respect to the standard basis is given by the matrix $A=\begin{bmatrix}
    1 & 0 & 2 \\
    0 &3 &0 \\
    4 & 0 & 5
    \end{bmatrix}$.
    (a) Prove that the linear transformation $T$ sends points on the $x$-$z$ plane to points on the $x$-$z$ plane.
    (b) Prove that the restriction of $T$ on the $x$-$z$ plane is a linear transformation.
    (c) Find the matrix representation of the linear transformation obtained in part (b) with respect to the standard basis $\left\{\, \begin{bmatrix}
    1 \\
    0 \\
    0
    \end{bmatrix}, \begin{bmatrix}
    0 \\
    0 \\
    1
    \end{bmatrix} \,\right\}$ of the $x$-$z$ plane.

  30. Let $T:\R^3 \to \R^3$ be the linear transformation defined by the formula
    \[T\left(\, \begin{bmatrix}
    x_1 \\
    x_2 \\
    x_3
    \end{bmatrix} \,\right)=\begin{bmatrix}
    x_1+3x_2-2x_3 \\
    2x_1+3x_2 \\
    x_2+x_3
    \end{bmatrix}.\] Determine whether $T$ is an isomorphism and if so find the formula for the inverse linear transformation $T^{-1}$.

  31. Let $\R^n$ be an inner product space with inner product $\langle \mathbf{x}, \mathbf{y}\rangle=\mathbf{x}^{\trans}\mathbf{y}$ for $\mathbf{x}, \mathbf{y}\in \R^n$. A linear transformation $T:\R^n \to \R^n$ is called orthogonal transformation if for all $\mathbf{x}, \mathbf{y}\in \R^n$, it satisfies
    \[\langle T(\mathbf{x}), T(\mathbf{y})\rangle=\langle\mathbf{x}, \mathbf{y} \rangle.\] Prove that if $T:\R^n\to \R^n$ is an orthogonal transformation, then $T$ is an isomorphism.