Can a Linear Transformation From R1 to R 2 Be Onto? Why?
Linear Transformation from $\R^due north$ to $\R^m$
Linear Transformation from $\R^n$ to $\R^m$
Definition
- 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 accept
- $T(\mathbf{x}+\mathbf{y})=T(\mathbf{x})+T(\mathbf{y})$
- $T(c\mathbf{x})=cT(\mathbf{x})$
- The nullspace $\calN(T)$ of a linear transformation $T:\R^northward\to \R^m$ is
\[\calN(T)=\{\mathbf{10}\in \R^north \mid T(\mathbf{x})=\mathbf{0}_m\}.\] - The nullity of $T$ is the dimension of $\calN(T)$.
- The range $\calR(T)$ of a linear transformation $T:\R^n\to \R^m$ is
\[\calR(T)=\{\mathbf{y}\in \R^m \mid \mathbf{y}=T(\mathbf{x}) \text{ for some } \mathbf{x}\in \R^n\}.\] - The rank of $T$ is the dimension of $\calR(T)$.
- The matrix representation of a linear transformation $T:\R^northward \to \R^m$ is an $m\times northward$ matrix $A$ such that $T(\mathbf{x})=A\mathbf{x}$ for all $\mathbf{ten}\in \R^n$.
Summary
Allow $T:\R^n \to \R^m$ be a linear transformation.
- $T(\mathbf{0}_n)=\mathbf{0}_m$, where $\mathbf{0}_n$ and $\mathbf{0}_m$ are the zero vectors in $\R^n$ and $R^chiliad$, respectively.
- The matrix representation $A$ of a linear transformation $T:\R^north \to \R^m$ is given by $A=[T(\mathbf{e}_1), \dots, T(\mathbf{e}_n)]$, where $\mathbf{east}_1, \dots, \mathbf{due east}_n$ are the standard basis for $\R^due north$.
- If $A$ is the matrix representaiton of a linear transformation $T$, then
- $\calN(T)=\calN(A)$ and $\calR(T)=\calR(A)$.
- The nullity of $T$ is the same as the nullity of $A$.
- The rank of $T$ is the aforementioned equally the rank of $A$.
=solution
Bug
- Define two functions $T:\R^{2}\to\R^{two}$ and $Due south:\R^{2}\to\R^{two}$ by
\[
T\left(
\brainstorm{bmatrix}
ten \\ y
\end{bmatrix}
\right)
=
\begin{bmatrix}
2x+y \\ 0
\end{bmatrix}
,\;
S\left(
\begin{bmatrix}
x \\ y
\cease{bmatrix}
\right)
=
\begin{bmatrix}
x+y \\ xy
\terminate{bmatrix}
.
\] Determine whether $T$, $S$, and the blended $S\circ T$ are linear transformations. - Allow $T:\R^2 \to \R^3$ 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 $\R^two$ and
\[\mathbf{u}_1=\brainstorm{bmatrix}
5 \\
1 \\
2
\finish{bmatrix} \text{ and } \mathbf{u}_2=\brainstorm{bmatrix}
8 \\
2 \\
6
\end{bmatrix}.\] Then observe $T\left(\, \brainstorm{bmatrix}
3 \\
-two
\end{bmatrix} \,\right)$.See (b)
- Let $T : \mathbb{R}^northward \to \mathbb{R}^one thousand$ be a linear transformation. Let $\mathbf{0}_n$ and $\mathbf{0}_m$ be zero vectors of $\mathbb{R}^due north$ and $\mathbb{R}^m$, respectively. Bear witness that $T(\mathbf{0}_n)=\mathbf{0}_m$.
- Make up one's mind whether the function $T:\R^2 \to \R^3$ divers by
\[T\left(\, \begin{bmatrix}
x \\
y
\finish{bmatrix} \,\right)
=
\brainstorm{bmatrix}
x_+y \\
x+1 \\
3y
\end{bmatrix}\] is a linear transformation. - Allow $T: \R^two \to \R^2$ be a linear transformation.
Allow
\[
\mathbf{u}=\begin{bmatrix}
1 \\
two
\stop{bmatrix}, \mathbf{v}=\begin{bmatrix}
3 \\
five
\end{bmatrix}\] be 2-dimensional vectors. Suppose that
\begin{align*}
T(\mathbf{u})&=T\left( \begin{bmatrix}
1 \\
2
\end{bmatrix} \right)=\brainstorm{bmatrix}
-iii \\
5
\end{bmatrix},\\
T(\mathbf{5})&=T\left(\brainstorm{bmatrix}
iii \\
5
\terminate{bmatrix}\correct)=\begin{bmatrix}
7 \\
1
\end{bmatrix}.
\end{align*}
Permit $\mathbf{west}=\brainstorm{bmatrix}
10 \\
y
\end{bmatrix}\in \R^2$. Detect the formula for $T(\mathbf{westward})$ in terms of $x$ and $y$. - Let $\{\mathbf{v}_1, \mathbf{v}_2\}$ be a ground of the vector space $\R^ii$, where $\mathbf{five}_1=\brainstorm{bmatrix}
1 \\
1
\cease{bmatrix}$ and $\mathbf{five}_2=\begin{bmatrix}
one \\
-1
\end{bmatrix}$. The action of a linear transformation $T:\R^2\to \R^3$ on the basis $\{\mathbf{v}_1, \mathbf{5}_2\}$ is given by
\begin{align*}
T(\mathbf{five}_1)=\begin{bmatrix}
ii \\
iv \\
half dozen
\finish{bmatrix} \text{ and } T(\mathbf{five}_2)=\begin{bmatrix}
0 \\
8 \\
ten
\end{bmatrix}.
\stop{align*}
Find the formula of $T(\mathbf{x})$, where $\mathbf{x}=\brainstorm{bmatrix}
x \\
y
\stop{bmatrix}\in \R^2$. - Let $T: \R^iii \to \R^two$ be a linear transformation such that
\[T(\mathbf{e}_1)=\brainstorm{bmatrix}
ane \\
4
\end{bmatrix}, T(\mathbf{e}_2)=\begin{bmatrix}
2 \\
5
\end{bmatrix}, T(\mathbf{e}_3)=\begin{bmatrix}
3 \\
6
\cease{bmatrix},\] where
\[\mathbf{e}_1=\begin{bmatrix}
1 \\
0 \\
0
\end{bmatrix}, \mathbf{e}_2=\begin{bmatrix}
0 \\
1 \\
0
\cease{bmatrix}, \mathbf{e}_3=\begin{bmatrix}
0 \\
0 \\
1
\end{bmatrix}\] are the standard unit of measurement basis vectors of $\R^3$.
For any vector $\mathbf{ten}=\brainstorm{bmatrix}
x_1 \\
x_2 \\
x_3
\cease{bmatrix}\in \R^3$, 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 effigy below.
Find the matrix representation $A$ of the linear transformation $T$.
- Let $T$ be a linear transformation from $\R^iii$ to $\R^2$ such that
\[ T\left(\, \begin{bmatrix}
0 \\
1 \\
0
\stop{bmatrix}\,\right) =\begin{bmatrix}
1 \\
ii
\finish{bmatrix} \text{ and }T\left(\, \begin{bmatrix}
0 \\
1 \\
1
\stop{bmatrix}\,\correct)=\begin{bmatrix}
0 \\
1
\end{bmatrix}. \] Then detect $T\left(\, \brainstorm{bmatrix}
0 \\
1 \\
two
\cease{bmatrix} \,\right)$.
(The Ohio State University) - Allow $T:\R^3 \to \R^2$ exist a linear transformation such that
\[ T(\mathbf{e}_1)=\begin{bmatrix}
one \\
0
\end{bmatrix}, T(\mathbf{eastward}_2)=\begin{bmatrix}
0 \\
1
\stop{bmatrix}, T(\mathbf{e}_3)=\brainstorm{bmatrix}
one \\
0
\end{bmatrix},\] where $\mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3$ are the standard basis of $\R^iii$.
And then find the rank and the nullity of $T$.
(The Ohio State University) - Permit $T:\R^2 \to \R^iii$ be a linear transformation such that
\[T\left(\, \begin{bmatrix}
3 \\
2
\finish{bmatrix} \,\right)
=\begin{bmatrix}
1 \\
2 \\
3
\stop{bmatrix} \text{ and }
T\left(\, \brainstorm{bmatrix}
4\\
3
\terminate{bmatrix} \,\right)
=\begin{bmatrix}
0 \\
-5 \\
1
\cease{bmatrix}.\] (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 State Academy) - Define the map $T:\R^2 \to \R^iii$ by $T \left ( \begin{bmatrix}
x_1 \\
x_2
\terminate{bmatrix}\correct )=\begin{bmatrix}
x_1-x_2 \\
x_1+x_2 \\
x_2
\end{bmatrix}$.
(a) Show that $T$ is a linear transformation.
(b) Discover a matrix $A$ such that $T(\mathbf{10})=A\mathbf{x}$ 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 $T:\R^4 \to \R^3$ be a linear transformation divers past
\[ T\left (\, \brainstorm{bmatrix}
x_1 \\
x_2 \\
x_3 \\
x_4
\end{bmatrix} \,\right) = \begin{bmatrix}
x_1+2x_2+3x_3-x_4 \\
3x_1+5x_2+8x_3-2x_4 \\
x_1+x_2+2x_3
\end{bmatrix}.\] (a) Notice a matrix $A$ such that $T(\mathbf{x})=A\mathbf{x}$.
(b) Find a footing for the null space of $T$.
(c) Find the rank of the linear transformation $T$.
(The Ohio State Academy) - Let $T:\R^2 \to \R^3$ be a linear transformation such that $T(\mathbf{e}_1)=\mathbf{u}_1$ and $T(\mathbf{due east}_2)=\mathbf{u}_2$, where $\mathbf{e}_1=\begin{bmatrix}
i \\
0
\terminate{bmatrix}, \mathbf{e}_2=\begin{bmatrix}
0 \\
i
\end{bmatrix}$ are unit vectors of $\R^2$ and
\[\mathbf{u}_1= \begin{bmatrix}
-1 \\
0 \\
1
\end{bmatrix}, \quad \mathbf{u}_2=\brainstorm{bmatrix}
two \\
1 \\
0
\end{bmatrix}.\] Then find $T\left(\begin{bmatrix}
iii \\
-2
\end{bmatrix}\right)$. - Let $B=\{\mathbf{v}_1, \mathbf{five}_2 \}$ be a footing for the vector space $\R^ii$, and let $T:\R^2 \to \R^ii$ exist a linear transformation such that
\[T(\mathbf{v}_1)=\begin{bmatrix}
i \\
-2
\end{bmatrix} \text{ and } T(\mathbf{v}_2)=\begin{bmatrix}
iii \\
one
\end{bmatrix}.\] If $\mathbf{e}_1=\mathbf{v}_1+2\mathbf{v}_2 \text{ and } \mathbf{e}_2=2\mathbf{five}_1-\mathbf{u}_2$, where $\mathbf{east}_1, \mathbf{eastward}_2$ are the standard unit of measurement vectors in $\R^2$, then detect the matrix of $T$ with respect to the basis $\{\mathbf{east}_1, \mathbf{e}_2\}$. - Permit $T: \R^2 \to \R^two$ be a linear transformation such that
\[T\left(\, \begin{bmatrix}
1 \\
one
\end{bmatrix} \,\correct)=\brainstorm{bmatrix}
4 \\
1
\end{bmatrix}, T\left(\, \begin{bmatrix}
0 \\
ane
\end{bmatrix} \,\right)=\begin{bmatrix}
3 \\
2
\end{bmatrix}.\] So 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) - If $L:\R^2 \to \R^3$ is a linear transformation such that
\begin{align*}
L\left( \begin{bmatrix}
i \\
0
\end{bmatrix}\right)
=\begin{bmatrix}
i \\
1 \\
2
\end{bmatrix}, \,\,\,\,
50\left( \brainstorm{bmatrix}
1 \\
i
\end{bmatrix}\right)
=\begin{bmatrix}
2 \\
3 \\
ii
\end{bmatrix}.
\end{marshal*}
and then
(a) find $Fifty\left( \begin{bmatrix}
1 \\
2
\end{bmatrix}\right)$, and
(b) observe the formula for $L\left( \begin{bmatrix}
x \\
y
\cease{bmatrix}\correct)$.
(Purdue Academy) - Suppose that $T: \R^2 \to \R^iii$ is a linear transformation satisfying
\[T\left(\, \begin{bmatrix}
i \\
two
\end{bmatrix}\,\right)=\begin{bmatrix}
iii \\
4 \\
five
\finish{bmatrix} \text{ and } T\left(\, \begin{bmatrix}
0 \\
i
\cease{bmatrix} \,\correct)=\begin{bmatrix}
0 \\
0 \\
1
\finish{bmatrix}.\] Find 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 $iii$-dimensional vector space $\R^three$ to $\R^3$ itself satisfying the following relations.
\begin{marshal*}
T\left(\, \brainstorm{bmatrix}
1 \\
ane \\
ane
\end{bmatrix} \,\right)
=\begin{bmatrix}
i \\
0 \\
1
\stop{bmatrix}, \qquad T\left(\, \begin{bmatrix}
2 \\
iii \\
5
\terminate{bmatrix} \, \right) =
\begin{bmatrix}
0 \\
2 \\
-1
\finish{bmatrix}, \qquad
T \left( \, \begin{bmatrix}
0 \\
1 \\
2
\cease{bmatrix} \, \right)=
\brainstorm{bmatrix}
1 \\
0 \\
0
\end{bmatrix}.
\end{marshal*}
Then for whatsoever vector $\mathbf{x}=\begin{bmatrix}
ten \\
y \\
z
\cease{bmatrix}\in \R^3$, find the formula for $T(\mathbf{ten})$. - Let $T:\R^2 \to \R^iii$ be a linear transformation given by
\[T\left(\, \begin{bmatrix}
x_1 \\
x_2
\stop{bmatrix} \,\right)
=
\begin{bmatrix}
x_1-x_2 \\
x_2 \\
x_1+ x_2
\end{bmatrix}.\] Notice an orthonormal basis of the range of $T$.
(The Ohio Land Academy) - Allow $T: \R^northward \to \R^m$ be a linear transformation. Suppose that $Southward=\{\mathbf{x}_1, \mathbf{x}_2,\dots, \mathbf{x}_k\}$ is a subset of $\R^n$ such that $\{T(\mathbf{ten}_1), T(\mathbf{ten}_2), \dots, T(\mathbf{x}_k) \}$ is a linearly independent subset of $\R^m$. Prove that the gear up $South$ is linearly contained.
- Permit $T: \R^north \to \R^one thousand$ exist a linear transformation. Suppose that the nullity of $T$ is nil. If $\{\mathbf{x}_1, \mathbf{x}_2,\dots, \mathbf{x}_k\}$ is a linearly independent subset of $\R^n$, then prove that $\{T(\mathbf{ten}_1), T(\mathbf{x}_2), \dots, T(\mathbf{x}_k) \}$ is a linearly independent subset of $\R^m$.
- Let $T$ be a linear transformation from the vector space $\R^3$ to $\R^three$.
Suppose that $k=3$ is the smallest positive integer such that $T^k=\mathbf{0}$ (the zero linear transformation) and suppose that we take $\mathbf{x}\in \R^3$ such that $T^2\mathbf{x}\neq \mathbf{0}$. Show that the vectors $\mathbf{x}, T\mathbf{ten}, T^two\mathbf{x}$ course a basis for $\R^iii$.
(The Ohio State Academy) - Permit $n$ be a positive integer. Let $T:\R^north \to \R$ be a not-zilch linear transformation. Evidence the followings.
(a) The nullity of $T$ is $n-ane$. That is, the dimension of the nullspace of $T$ is $n-1$.
(b) Permit $B=\{\mathbf{5}_1, \cdots, \mathbf{v}_{n-ane}\}$ be a basis of the nullspace $\calN(T)$ of $T$. Let $\mathbf{west}$ be the $n$-dimensional vector that is not in $\calN(T)$. Then $B'=\{\mathbf{v}_1, \cdots, \mathbf{v}_{n-one}, \mathbf{westward}\}$ is a ground of $\R^north$.
(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 $\mathbf{five}\in \calN(T)$. - Allow $V$ 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 $A$. - Let $\mathbf{u}=\brainstorm{bmatrix}
ane \\
1 \\
0
\terminate{bmatrix}$ 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 goose egg infinite $\calN(T)$, a footing for $\calN(T)$ and nullity of $T$.
(b) Only by using role (a) and no other calculations, find $\det(A)$, where $A$ is the matrix representation of $T$ with respect to the standard ground of $\R^three$.
(c) Calculate the range $\calR(T)$, a footing for $\calR(T)$ and the rank of $T$.
(d) Calculate the matrix $A$ representing $T$ with respect to the standard footing for $\R^iii$.
(e) Permit $B=\left\{\, \brainstorm{bmatrix}
1 \\
0 \\
0
\end{bmatrix}, \begin{bmatrix}
-i \\
ane \\
0
\end{bmatrix}, \brainstorm{bmatrix}
0 \\
-1 \\
i
\end{bmatrix} \,\right\}$ be a basis for $\R^iii$. 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 $T:\R^iii\to \R^3$ by
\[T(\mathbf{v})=\mathbf{a}\times \mathbf{five}\] for all $\mathbf{v}\in \R^3$. Here the right-hand side is the cross product of $\mathbf{a}$ and $\mathbf{5}$.
(a) Testify that $T:\R^3\to \R^3$ is a linear transformation.
(b) Decide the eigenvalues and eigenvectors of $T$. - Determine all linear transformations of the $2$-dimensional $x$-$y$ aeroplane $\R^ii$ that take the line $y=x$ to the line $y=-x$.
- Let $F:\R^2\to \R^ii$ be the function that maps each vector in $\R^2$ to its reflection with respect to $10$-axis. Determine 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 $\R^2$ (the $x$-$y$-plane) to itself which is the reflection across a line $y=mx$ for some $m\in \R$. And then find the matrix representation of the linear transformation $T$ with respect to the standard footing $B=\{\mathbf{east}_1, \mathbf{east}_2\}$ of $\R^2$, where $\mathbf{e}_1=\begin{bmatrix}
1 \\
0
\cease{bmatrix}, \mathbf{e}_2=\brainstorm{bmatrix}
0 \\
1
\cease{bmatrix}$. - Let $T:\R^iii \to \R^three$ be a linear transformation and suppose that its matrix representation with respect to the standard basis is given past the matrix $A=\begin{bmatrix}
1 & 0 & 2 \\
0 &3 &0 \\
4 & 0 & v
\finish{bmatrix}$.
(a) Prove that the linear transformation $T$ sends points on the $x$-$z$ aeroplane to points on the $ten$-$z$ aeroplane.
(b) Prove that the brake of $T$ on the $x$-$z$ plane is a linear transformation.
(c) Discover the matrix representation of the linear transformation obtained in part (b) with respect to the standard basis $\left\{\, \begin{bmatrix}
ane \\
0 \\
0
\end{bmatrix}, \begin{bmatrix}
0 \\
0 \\
1
\cease{bmatrix} \,\right\}$ of the $10$-$z$ plane. - Allow $T:\R^three \to \R^3$ be the linear transformation divers by the formula
\[T\left(\, \begin{bmatrix}
x_1 \\
x_2 \\
x_3
\end{bmatrix} \,\right)=\begin{bmatrix}
x_1+3x_2-2x_3 \\
2x_1+3x_2 \\
x_2+x_3
\end{bmatrix}.\] Decide whether $T$ is an isomorphism and if and so notice the formula for the inverse linear transformation $T^{-1}$. - Let $\R^n$ exist an inner product space with inner production $\langle \mathbf{10}, \mathbf{y}\rangle=\mathbf{x}^{\trans}\mathbf{y}$ for $\mathbf{x}, \mathbf{y}\in \R^n$. A linear transformation $T:\R^n \to \R^n$ is chosen orthogonal transformation if for all $\mathbf{10}, \mathbf{y}\in \R^due north$, information technology satisfies
\[\langle T(\mathbf{10}), T(\mathbf{y})\rangle=\langle\mathbf{ten}, \mathbf{y} \rangle.\] Show that if $T:\R^n\to \R^n$ is an orthogonal transformation, so $T$ is an isomorphism.
Source: https://yutsumura.com/linear-algebra/linear-transformation-from-rn-to-rm/
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