Suppose F:rn to Rm Is a Linear Map. What Is the Derivative of F
Linear Transformation from $\R^north$ to $\R^thousand$
Linear Transformation from $\R^n$ to $\R^grand$
Definition
- A function $T:\R^n \to \R^m$ is called a linear transformation if $T$ satisfies the post-obit 2 linearity atmospheric condition: For any $\mathbf{x}, \mathbf{y}\in \R^n$ and $c\in \R$, nosotros have
- $T(\mathbf{x}+\mathbf{y})=T(\mathbf{10})+T(\mathbf{y})$
- $T(c\mathbf{x})=cT(\mathbf{x})$
- The nullspace $\calN(T)$ of a linear transformation $T:\R^n\to \R^thousand$ is
\[\calN(T)=\{\mathbf{x}\in \R^n \mid T(\mathbf{ten})=\mathbf{0}_m\}.\] - The nullity of $T$ is the dimension of $\calN(T)$.
- The range $\calR(T)$ of a linear transformation $T:\R^n\to \R^thousand$ is
\[\calR(T)=\{\mathbf{y}\in \R^thou \mid \mathbf{y}=T(\mathbf{ten}) \text{ for some } \mathbf{x}\in \R^n\}.\] - The rank of $T$ is the dimension of $\calR(T)$.
- The matrix representation of a linear transformation $T:\R^due north \to \R^m$ is an $thou\times n$ matrix $A$ such that $T(\mathbf{x})=A\mathbf{x}$ for all $\mathbf{x}\in \R^n$.
Summary
Let $T:\R^n \to \R^k$ be a linear transformation.
- $T(\mathbf{0}_n)=\mathbf{0}_m$, where $\mathbf{0}_n$ and $\mathbf{0}_m$ are the naught vectors in $\R^northward$ and $R^yard$, respectively.
- The matrix representation $A$ of a linear transformation $T:\R^n \to \R^m$ is given by $A=[T(\mathbf{due east}_1), \dots, T(\mathbf{e}_n)]$, where $\mathbf{e}_1, \dots, \mathbf{eastward}_n$ are the standard ground for $\R^due north$.
- If $A$ is the matrix representaiton of a linear transformation $T$, so
- $\calN(T)=\calN(A)$ and $\calR(T)=\calR(A)$.
- The nullity of $T$ is the same every bit the nullity of $A$.
- The rank of $T$ is the same as the rank of $A$.
=solution
Problems
- Ascertain ii functions $T:\R^{ii}\to\R^{2}$ and $S:\R^{ii}\to\R^{ii}$ by
\[
T\left(
\begin{bmatrix}
x \\ y
\end{bmatrix}
\correct)
=
\begin{bmatrix}
2x+y \\ 0
\finish{bmatrix}
,\;
S\left(
\brainstorm{bmatrix}
x \\ y
\stop{bmatrix}
\correct)
=
\begin{bmatrix}
x+y \\ xy
\stop{bmatrix}
.
\] Decide whether $T$, $S$, and the composite $Due south\circ T$ are linear transformations. - Let $T:\R^2 \to \R^three$ 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
\[\mathbf{u}_1=\begin{bmatrix}
5 \\
1 \\
2
\terminate{bmatrix} \text{ and } \mathbf{u}_2=\begin{bmatrix}
8 \\
2 \\
6
\finish{bmatrix}.\] Then detect $T\left(\, \begin{bmatrix}
three \\
-2
\finish{bmatrix} \,\right)$.See (b)
- Let $T : \mathbb{R}^n \to \mathbb{R}^yard$ exist a linear transformation. Permit $\mathbf{0}_n$ and $\mathbf{0}_m$ be zero vectors of $\mathbb{R}^n$ and $\mathbb{R}^chiliad$, respectively. Show that $T(\mathbf{0}_n)=\mathbf{0}_m$.
- Make up one's mind whether the function $T:\R^ii \to \R^3$ defined past
\[T\left(\, \brainstorm{bmatrix}
x \\
y
\end{bmatrix} \,\right)
=
\begin{bmatrix}
x_+y \\
x+1 \\
3y
\terminate{bmatrix}\] is a linear transformation. - Let $T: \R^2 \to \R^2$ be a linear transformation.
Allow
\[
\mathbf{u}=\brainstorm{bmatrix}
1 \\
2
\end{bmatrix}, \mathbf{v}=\begin{bmatrix}
3 \\
5
\terminate{bmatrix}\] be ii-dimensional vectors. Suppose that
\begin{marshal*}
T(\mathbf{u})&=T\left( \brainstorm{bmatrix}
1 \\
2
\terminate{bmatrix} \right)=\begin{bmatrix}
-3 \\
5
\cease{bmatrix},\\
T(\mathbf{v})&=T\left(\begin{bmatrix}
3 \\
5
\end{bmatrix}\correct)=\brainstorm{bmatrix}
7 \\
1
\end{bmatrix}.
\end{align*}
Let $\mathbf{w}=\begin{bmatrix}
x \\
y
\end{bmatrix}\in \R^2$. Observe the formula for $T(\mathbf{w})$ in terms of $x$ and $y$. - Permit $\{\mathbf{v}_1, \mathbf{v}_2\}$ be a footing of the vector space $\R^2$, where $\mathbf{five}_1=\begin{bmatrix}
1 \\
1
\stop{bmatrix}$ and $\mathbf{v}_2=\begin{bmatrix}
i \\
-i
\end{bmatrix}$. 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{five}_1)=\begin{bmatrix}
ii \\
four \\
6
\end{bmatrix} \text{ and } T(\mathbf{v}_2)=\begin{bmatrix}
0 \\
viii \\
10
\end{bmatrix}.
\end{marshal*}
Detect the formula of $T(\mathbf{ten})$, where $\mathbf{x}=\begin{bmatrix}
10 \\
y
\end{bmatrix}\in \R^2$. - Let $T: \R^3 \to \R^2$ be a linear transformation such that
\[T(\mathbf{e}_1)=\begin{bmatrix}
one \\
iv
\terminate{bmatrix}, T(\mathbf{e}_2)=\brainstorm{bmatrix}
2 \\
v
\end{bmatrix}, T(\mathbf{e}_3)=\begin{bmatrix}
3 \\
6
\end{bmatrix},\] where
\[\mathbf{e}_1=\begin{bmatrix}
i \\
0 \\
0
\end{bmatrix}, \mathbf{e}_2=\begin{bmatrix}
0 \\
1 \\
0
\end{bmatrix}, \mathbf{e}_3=\begin{bmatrix}
0 \\
0 \\
one
\end{bmatrix}\] 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 $T(\mathbf{x})$. - Allow $T:\R^2\to \R^2$ be a linear transformation such that it maps the vectors $\mathbf{v}_1, \mathbf{v}_2$ as indicated in the figure beneath.
Discover the matrix representation $A$ of the linear transformation $T$.
- Let $T$ be a linear transformation from $\R^3$ to $\R^2$ such that
\[ T\left(\, \begin{bmatrix}
0 \\
i \\
0
\end{bmatrix}\,\correct) =\begin{bmatrix}
i \\
2
\end{bmatrix} \text{ and }T\left(\, \brainstorm{bmatrix}
0 \\
1 \\
1
\stop{bmatrix}\,\right)=\begin{bmatrix}
0 \\
1
\finish{bmatrix}. \] Then find $T\left(\, \begin{bmatrix}
0 \\
1 \\
2
\end{bmatrix} \,\right)$.
(The Ohio Country University) - Allow $T:\R^three \to \R^2$ exist a linear transformation such that
\[ T(\mathbf{e}_1)=\begin{bmatrix}
ane \\
0
\cease{bmatrix}, T(\mathbf{e}_2)=\begin{bmatrix}
0 \\
1
\end{bmatrix}, T(\mathbf{e}_3)=\begin{bmatrix}
1 \\
0
\end{bmatrix},\] where $\mathbf{e}_1, \mathbf{e}_2, \mathbf{due east}_3$ are the standard basis of $\R^3$.
Then discover the rank and the nullity of $T$.
(The Ohio State University) - Let $T:\R^2 \to \R^3$ be a linear transformation such that
\[T\left(\, \begin{bmatrix}
3 \\
two
\end{bmatrix} \,\right)
=\brainstorm{bmatrix}
one \\
two \\
3
\end{bmatrix} \text{ and }
T\left(\, \begin{bmatrix}
4\\
3
\stop{bmatrix} \,\correct)
=\begin{bmatrix}
0 \\
-5 \\
i
\cease{bmatrix}.\] (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 Land University) - Ascertain the map $T:\R^2 \to \R^3$ past $T \left ( \brainstorm{bmatrix}
x_1 \\
x_2
\end{bmatrix}\right )=\begin{bmatrix}
x_1-x_2 \\
x_1+x_2 \\
x_2
\end{bmatrix}$.
(a) Prove 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 infinite (kernel) and the range of $T$ and give the rank and the nullity of $T$. - Let $T:\R^iv \to \R^3$ exist a linear transformation divers past
\[ T\left (\, \begin{bmatrix}
x_1 \\
x_2 \\
x_3 \\
x_4
\finish{bmatrix} \,\correct) = \begin{bmatrix}
x_1+2x_2+3x_3-x_4 \\
3x_1+5x_2+8x_3-2x_4 \\
x_1+x_2+2x_3
\stop{bmatrix}.\] (a) Find a matrix $A$ such that $T(\mathbf{ten})=A\mathbf{x}$.
(b) Find a basis for the null space of $T$.
(c) Discover the rank of the linear transformation $T$.
(The Ohio State University) - Let $T:\R^ii \to \R^3$ be a linear transformation such that $T(\mathbf{e}_1)=\mathbf{u}_1$ and $T(\mathbf{e}_2)=\mathbf{u}_2$, where $\mathbf{east}_1=\brainstorm{bmatrix}
ane \\
0
\end{bmatrix}, \mathbf{e}_2=\brainstorm{bmatrix}
0 \\
1
\end{bmatrix}$ are unit of measurement vectors of $\R^ii$ and
\[\mathbf{u}_1= \brainstorm{bmatrix}
-1 \\
0 \\
1
\end{bmatrix}, \quad \mathbf{u}_2=\brainstorm{bmatrix}
2 \\
1 \\
0
\finish{bmatrix}.\] So find $T\left(\begin{bmatrix}
3 \\
-2
\end{bmatrix}\right)$. - Let $B=\{\mathbf{v}_1, \mathbf{v}_2 \}$ be a ground for the vector infinite $\R^ii$, and let $T:\R^two \to \R^2$ be a linear transformation such that
\[T(\mathbf{v}_1)=\begin{bmatrix}
one \\
-2
\finish{bmatrix} \text{ and } T(\mathbf{v}_2)=\begin{bmatrix}
3 \\
ane
\finish{bmatrix}.\] If $\mathbf{e}_1=\mathbf{5}_1+2\mathbf{5}_2 \text{ and } \mathbf{e}_2=two\mathbf{v}_1-\mathbf{u}_2$, where $\mathbf{e}_1, \mathbf{e}_2$ are the standard unit vectors in $\R^two$, then find the matrix of $T$ with respect to the basis $\{\mathbf{east}_1, \mathbf{e}_2\}$. - Allow $T: \R^2 \to \R^two$ exist a linear transformation such that
\[T\left(\, \begin{bmatrix}
1 \\
i
\end{bmatrix} \,\right)=\brainstorm{bmatrix}
4 \\
1
\end{bmatrix}, T\left(\, \begin{bmatrix}
0 \\
1
\end{bmatrix} \,\right)=\begin{bmatrix}
3 \\
2
\end{bmatrix}.\] And then discover the matrix $A$ such that $T(\mathbf{10})=A\mathbf{x}$ for every $\mathbf{x}\in \R^2$, and find the rank and nullity of $T$.
(The Ohio State University) - If $Fifty:\R^2 \to \R^iii$ is a linear transformation such that
\begin{marshal*}
L\left( \brainstorm{bmatrix}
ane \\
0
\cease{bmatrix}\right)
=\brainstorm{bmatrix}
1 \\
1 \\
2
\finish{bmatrix}, \,\,\,\,
50\left( \begin{bmatrix}
ane \\
1
\stop{bmatrix}\correct)
=\begin{bmatrix}
2 \\
three \\
2
\finish{bmatrix}.
\cease{align*}
and then
(a) find $L\left( \begin{bmatrix}
1 \\
two
\finish{bmatrix}\right)$, and
(b) find the formula for $L\left( \begin{bmatrix}
x \\
y
\end{bmatrix}\right)$.
(Purdue University) - Suppose that $T: \R^2 \to \R^3$ is a linear transformation satisfying
\[T\left(\, \begin{bmatrix}
1 \\
2
\finish{bmatrix}\,\correct)=\begin{bmatrix}
three \\
4 \\
5
\end{bmatrix} \text{ and } T\left(\, \begin{bmatrix}
0 \\
1
\end{bmatrix} \,\correct)=\begin{bmatrix}
0 \\
0 \\
1
\end{bmatrix}.\] Detect a full general formula for $T\left(\, \brainstorm{bmatrix}
x_1 \\
x_2
\end{bmatrix} \,\right)$.
(The Ohio State University) - Allow $T$ be the linear transformation from the $3$-dimensional vector space $\R^three$ to $\R^3$ itself satisfying the post-obit relations.
\begin{align*}
T\left(\, \begin{bmatrix}
i \\
1 \\
1
\cease{bmatrix} \,\right)
=\begin{bmatrix}
ane \\
0 \\
i
\end{bmatrix}, \qquad T\left(\, \begin{bmatrix}
2 \\
3 \\
5
\finish{bmatrix} \, \right) =
\begin{bmatrix}
0 \\
2 \\
-ane
\terminate{bmatrix}, \qquad
T \left( \, \begin{bmatrix}
0 \\
1 \\
2
\end{bmatrix} \, \right)=
\brainstorm{bmatrix}
ane \\
0 \\
0
\terminate{bmatrix}.
\cease{align*}
Then for any vector $\mathbf{x}=\begin{bmatrix}
x \\
y \\
z
\terminate{bmatrix}\in \R^3$, find the formula for $T(\mathbf{x})$. - Let $T:\R^two \to \R^iii$ be a linear transformation given by
\[T\left(\, \begin{bmatrix}
x_1 \\
x_2
\finish{bmatrix} \,\right)
=
\begin{bmatrix}
x_1-x_2 \\
x_2 \\
x_1+ x_2
\end{bmatrix}.\] Find an orthonormal basis of the range of $T$.
(The Ohio Country University) - Allow $T: \R^due north \to \R^m$ be a linear transformation. Suppose that $South=\{\mathbf{x}_1, \mathbf{x}_2,\dots, \mathbf{x}_k\}$ 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 $\R^m$. Bear witness that the set $S$ is linearly independent.
- Let $T: \R^north \to \R^m$ be a linear transformation. Suppose that the nullity of $T$ is zero. If $\{\mathbf{10}_1, \mathbf{ten}_2,\dots, \mathbf{x}_k\}$ is a linearly independent subset of $\R^n$, then show that $\{T(\mathbf{x}_1), T(\mathbf{ten}_2), \dots, T(\mathbf{x}_k) \}$ is a linearly contained subset of $\R^one thousand$.
- Let $T$ be a linear transformation from the vector space $\R^3$ to $\R^3$.
Suppose that $k=iii$ is the smallest positive integer such that $T^k=\mathbf{0}$ (the naught linear transformation) and suppose that we have $\mathbf{x}\in \R^3$ such that $T^2\mathbf{10}\neq \mathbf{0}$. Prove that the vectors $\mathbf{x}, T\mathbf{x}, T^two\mathbf{x}$ form a basis for $\R^3$.
(The Ohio Country University) - Allow $n$ be a positive integer. Let $T:\R^n \to \R$ be a non-cypher linear transformation. Prove the followings.
(a) The nullity of $T$ is $due north-1$. That is, the dimension of the nullspace of $T$ is $due north-1$.
(b) Let $B=\{\mathbf{v}_1, \cdots, \mathbf{v}_{n-ane}\}$ be a basis of the nullspace $\calN(T)$ of $T$. Let $\mathbf{w}$ be the $due north$-dimensional vector that is not in $\calN(T)$. Then $B'=\{\mathbf{5}_1, \cdots, \mathbf{v}_{n-one}, \mathbf{w}\}$ is a footing of $\R^n$.
(c) Each vector $\mathbf{u}\in \R^northward$ can exist expressed as
\[\mathbf{u}=\mathbf{5}+\frac{T(\mathbf{u})}{T(\mathbf{west})}\mathbf{w}\] for some vector $\mathbf{v}\in \calN(T)$. - Let $Five$ be the subspace of $\R^iv$ defined past the equation
\[x_1-x_2+2x_3+6x_4=0.\] Detect 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 $A$. - Let $\mathbf{u}=\begin{bmatrix}
1 \\
1 \\
0
\finish{bmatrix}$ and $T:\R^3 \to \R^3$ exist 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) Summate the nil infinite $\calN(T)$, a basis for $\calN(T)$ and nullity of $T$.
(b) Only past using part (a) and no other calculations, find $\det(A)$, where $A$ is the matrix representation of $T$ with respect to the standard ground of $\R^3$.
(c) Summate 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 footing for $\R^3$.
(e) Permit $B=\left\{\, \begin{bmatrix}
1 \\
0 \\
0
\end{bmatrix}, \begin{bmatrix}
-1 \\
i \\
0
\finish{bmatrix}, \begin{bmatrix}
0 \\
-1 \\
1
\end{bmatrix} \,\correct\}$ be a basis for $\R^three$. Calculate the coordinates of $\begin{bmatrix}
x \\
y \\
z
\end{bmatrix}$ with respect to $B$.
(The Ohio State University) - We fix a nonzero vector $\mathbf{a}$ in $\R^iii$ and ascertain a map $T:\R^3\to \R^3$ by
\[T(\mathbf{v})=\mathbf{a}\times \mathbf{v}\] for all $\mathbf{v}\in \R^iii$. Here the right-hand side is the cross production 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 $T$. - Make up one's mind all linear transformations of the $2$-dimensional $x$-$y$ aeroplane $\R^two$ that take the line $y=x$ to the line $y=-x$.
- Let $F:\R^2\to \R^two$ exist the function that maps each vector in $\R^2$ to its reflection with respect to $x$-axis. Make up one's mind the formula for the function $F$ and prove that $F$ is a linear transformation.
- Permit $T:\R^2 \to \R^2$ be a linear transformation of the $2$-dimensional vector space $\R^ii$ (the $x$-$y$-plane) to itself which is the reflection beyond a line $y=mx$ for some $k\in \R$. Then find the matrix representation of the linear transformation $T$ with respect to the standard basis $B=\{\mathbf{e}_1, \mathbf{due east}_2\}$ of $\R^2$, where $\mathbf{east}_1=\begin{bmatrix}
1 \\
0
\finish{bmatrix}, \mathbf{eastward}_2=\begin{bmatrix}
0 \\
1
\terminate{bmatrix}$. - Let $T:\R^3 \to \R^3$ exist a linear transformation and suppose that its matrix representation with respect to the standard basis is given by the matrix $A=\begin{bmatrix}
1 & 0 & 2 \\
0 &3 &0 \\
four & 0 & 5
\end{bmatrix}$.
(a) Prove that the linear transformation $T$ sends points on the $10$-$z$ aeroplane to points on the $x$-$z$ plane.
(b) Prove that the brake of $T$ on the $x$-$z$ aeroplane is a linear transformation.
(c) Find the matrix representation of the linear transformation obtained in part (b) with respect to the standard basis $\left\{\, \brainstorm{bmatrix}
ane \\
0 \\
0
\cease{bmatrix}, \begin{bmatrix}
0 \\
0 \\
1
\finish{bmatrix} \,\right\}$ of the $x$-$z$ plane. - Let $T:\R^3 \to \R^3$ be the linear transformation defined past 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 changed linear transformation $T^{-1}$. - Let $\R^northward$ be an inner product space with inner product $\langle \mathbf{x}, \mathbf{y}\rangle=\mathbf{x}^{\trans}\mathbf{y}$ for $\mathbf{ten}, \mathbf{y}\in \R^north$. A linear transformation $T:\R^northward \to \R^n$ is called orthogonal transformation if for all $\mathbf{10}, \mathbf{y}\in \R^due north$, information technology 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.
Source: https://yutsumura.com/linear-algebra/linear-transformation-from-rn-to-rm/
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