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

Problem 368

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, Linear Algebra Exam Problem)
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Basis of Span in Vector Space of Polynomials of Degree 2 or Less

Problem 367

Let $P_2$ be the vector space of all polynomials of degree $2$ or less with real coefficients.
Let
\[S=\{1+x+2x^2, \quad x+2x^2, \quad -1, \quad x^2\}\] be the set of four vectors in $P_2$.

Then find a basis of the subspace $\Span(S)$ among the vectors in $S$.

(Linear Algebra Exam Problem, the Ohio State University)
 
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Determine Whether Trigonometry Functions $\sin^2(x), \cos^2(x), 1$ are Linearly Independent or Dependent

Problem 365

Let $f(x)=\sin^2(x)$, $g(x)=\cos^2(x)$, and $h(x)=1$. These are vectors in $C[-1, 1]$.
Determine whether the set $\{f(x), \, g(x), \, h(x)\}$ is linearly dependent or linearly independent.

(The Ohio State University, Linear Algebra Midterm Exam Problem)
 
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True or False Problems of Vector Spaces and Linear Transformations

Problem 364

These are True or False problems.
For each of the following statements, determine if it contains a wrong information or not.

  1. Let $A$ be a $5\times 3$ matrix. Then the range of $A$ is a subspace in $\R^3$.
  2. The function $f(x)=x^2+1$ is not in the vector space $C[-1,1]$ because $f(0)=1\neq 0$.
  3. Since we have $\sin(x+y)=\sin(x)+\sin(y)$, the function $\sin(x)$ is a linear transformation.
  4. The set
    \[\left\{\, \begin{bmatrix}
    1 \\
    0 \\
    0
    \end{bmatrix}, \begin{bmatrix}
    0 \\
    1 \\
    1
    \end{bmatrix} \,\right\}\] is an orthonormal set.

(Linear Algebra Exam Problem, The Ohio State University)

 
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Quiz 11. Find Eigenvalues and Eigenvectors/ Properties of Determinants

Problem 363

(a) Find all the eigenvalues and eigenvectors of the matrix
\[A=\begin{bmatrix}
3 & -2\\
6& -4
\end{bmatrix}.\]

(b) Let
\[A=\begin{bmatrix}
1 & 0 & 3 \\
4 &5 &6 \\
7 & 0 & 9
\end{bmatrix} \text{ and } B=\begin{bmatrix}
2 & 0 & 0 \\
0 & 3 &0 \\
0 & 0 & 4
\end{bmatrix}.\] Then find the value of
\[\det(A^2B^{-1}A^{-2}B^2).\] (For part (b) without computation, you may assume that $A$ and $B$ are invertible matrices.)

 
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Quiz 10. Find Orthogonal Basis / Find Value of Linear Transformation

Problem 356

(a) Let $S=\{\mathbf{v}_1, \mathbf{v}_2\}$ be the set of the following vectors in $\R^4$.
\[\mathbf{v}_1=\begin{bmatrix}
1 \\
0 \\
1 \\
0
\end{bmatrix} \text{ and } \mathbf{v}_2=\begin{bmatrix}
0 \\
1 \\
1 \\
0
\end{bmatrix}.\] Find an orthogonal basis of the subspace $\Span(S)$ of $\R^4$.

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

 
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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)

 
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Hyperplane in $n$-Dimensional Space Through Origin is a Subspace

Problem 352

A hyperplane in $n$-dimensional vector space $\R^n$ is defined to be the set of vectors
\[\begin{bmatrix}
x_1 \\
x_2 \\
\vdots \\
x_n
\end{bmatrix}\in \R^n\] satisfying the linear equation of the form
\[a_1x_1+a_2x_2+\cdots+a_nx_n=b,\] where $a_1, a_2, \dots, a_n$ (at least one of $a_1, a_2, \dots, a_n$ is nonzero) and $b$ are real numbers.
Here at least one of $a_1, a_2, \dots, a_n$ is nonzero.

Consider the hyperplane $P$ in $\R^n$ described by the linear equation
\[a_1x_1+a_2x_2+\cdots+a_nx_n=0,\] where $a_1, a_2, \dots, a_n$ are some fixed real numbers and not all of these are zero.
(The constant term $b$ is zero.)

Then prove that the hyperplane $P$ is a subspace of $R^{n}$ of dimension $n-1$.

 
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