Tagged: matrix for linear transformation

Restriction of a Linear Transformation on the x-z Plane is a Linear Transformation

Problem 428

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.

 
Read solution

LoadingAdd to solve later

Find Matrix Representation of Linear Transformation From $\R^2$ to $\R^2$

Problem 370

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, Linear Algebra Exam Problem)
 
Read solution

LoadingAdd to solve later

Rank and Nullity of Linear Transformation From $\R^3$ to $\R^2$

Problem 369

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, Linear Algebra Exam Problem)
 
Read solution

LoadingAdd to solve later

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

Problem 353

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, Linear Algebra Math 2568 Exam Problem)

 
Read solution

LoadingAdd to solve later

Linear Transformation $T(X)=AX-XA$ and Determinant of Matrix Representation

Problem 330

Let $V$ be the vector space of all $n\times n$ real matrices.
Let us fix a matrix $A\in V$.
Define a map $T: V\to V$ by
\[ T(X)=AX-XA\] for each $X\in V$.

(a) Prove that $T:V\to V$ is a linear transformation.

(b) Let $B$ be a basis of $V$. Let $P$ be the matrix representation of $T$ with respect to $B$. Find the determinant of $P$.

 
Read solution

LoadingAdd to solve later

Determine linear transformation using matrix representation

Problem 324

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 $T(\mathbf{x})$.

 
Read solution

LoadingAdd to solve later

Basis with Respect to Which the Matrix for Linear Transformation is Diagonal

Problem 315

Let $P_1$ be the vector space of all real polynomials of degree $1$ or less. Consider the linear transformation $T: P_1 \to P_1$ defined by
\[T(ax+b)=(3a+b)x+a+3,\] for any $ax+b\in P_1$.

(a) With respect to the basis $B=\{1, x\}$, find the matrix of the linear transformation $T$.

(b) Find a basis $B’$ of the vector space $P_1$ such that the matrix of $T$ with respect to $B’$ is a diagonal matrix.

(c) Express $f(x)=5x+3$ as a linear combination of basis vectors of $B’$.

 
Read solution

LoadingAdd to solve later

Matrix Representation of a Linear Transformation of Subspace of Sequences Satisfying Recurrence Relation

Problem 309

Let $V$ be a real vector space of all real sequences
\[(a_i)_{i=1}^{\infty}=(a_1, a_2, \dots).\] Let $U$ be the subspace of $V$ consisting of all real sequences that satisfy the linear recurrence relation $a_{k+2}-5a_{k+1}+3a_{k}=0$ for $k=1, 2, \dots$.

(a) Let
\begin{align*}
\mathbf{u}_1&=(1, 0, -3, -15, -66, \dots)\\
\mathbf{u}_2&=(0, 1, 5, 22, 95, \dots)
\end{align*}
be vectors in $U$. Prove that $\{\mathbf{u}_1, \mathbf{u}_2\}$ is a basis of $U$ and conclude that the dimension of $U$ is $2$.


(b) Let $T$ be a map from $U$ to $U$ defined by
\[T\big((a_1, a_2, \dots)\big)=(a_2, a_3, \dots). \] Verify that the map $T$ actually sends a vector $(a_i)_{i=1}^{\infty}\in V$ to a vector $T\big((a_i)_{i=1}^{\infty}\big)$ in $U$, and show that $T$ is a linear transformation from $U$ to $U$.


(c) With respect to the basis $\{\mathbf{u}_1, \mathbf{u}_2\}$ obtained in (a), find the matrix representation $A$ of the linear transformation $T:U \to U$ from (b).

Read solution

LoadingAdd to solve later

Range, Null Space, Rank, and Nullity of a Linear Transformation from $\R^2$ to $\R^3$

Problem 154

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 $T$.

 
Read solution

LoadingAdd to solve later