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- Non-Prime Ideal of Continuous Functions
Let $R$ be the ring of all continuous functions on the interval $[0,1]$.
Let $I$ be the set of functions $f(x)$ in $R$ such that $f(1/2)=f(1/3)=0$.
Show that the set $I$ is an ideal of $R$ but is not a prime ideal.
Proof.
We first show that $I$ is an ideal of […]
- Sequences Satisfying Linear Recurrence Relation Form a Subspace
Let $V$ be a real vector space of all real sequences
\[(a_i)_{i=1}^{\infty}=(a_1, a_2, \cdots).\]
Let $U$ be the subset of $V$ defined by
\[U=\{ (a_i)_{i=1}^{\infty} \in V \mid a_{k+2}-5a_{k+1}+3a_{k}=0, k=1, 2, \dots \}.\]
Prove that $U$ is a subspace of […]
- Find the Nullspace and Range of the Linear Transformation $T(f)(x) = f(x)-f(0)$
Let $C([-1, 1])$ denote the vector space of real-valued functions on the interval $[-1, 1]$. Define the vector subspace
\[W = \{ f \in C([-1, 1]) \mid f(0) = 0 \}.\]
Define the map $T : C([-1, 1]) \rightarrow W$ by $T(f)(x) = f(x) - f(0)$. Determine if $T$ is a linear map. If […]
- Compute $A^{10}\mathbf{v}$ Using Eigenvalues and Eigenvectors of the Matrix $A$
Let
\[A=\begin{bmatrix}
1 & -14 & 4 \\
-1 &6 &-2 \\
-2 & 24 & -7
\end{bmatrix} \quad \text{ and }\quad \mathbf{v}=\begin{bmatrix}
4 \\
-1 \\
-7
\end{bmatrix}.\]
Find $A^{10}\mathbf{v}$.
You may use the following information without proving […]
- If $A$ is a Skew-Symmetric Matrix, then $I+A$ is Nonsingular and $(I-A)(I+A)^{-1}$ is Orthogonal
Let $A$ be an $n\times n$ real skew-symmetric matrix.
(a) Prove that the matrices $I-A$ and $I+A$ are nonsingular.
(b) Prove that
\[B=(I-A)(I+A)^{-1}\]
is an orthogonal matrix.
Proof.
(a) Prove that the matrices $I-A$ and $I+A$ are nonsingular.
The […]
- Diagonalize a 2 by 2 Matrix $A$ and Calculate the Power $A^{100}$
Let
\[A=\begin{bmatrix}
1 & 2\\
4& 3
\end{bmatrix}.\]
(a) Find eigenvalues of the matrix $A$.
(b) Find eigenvectors for each eigenvalue of $A$.
(c) Diagonalize the matrix $A$. That is, find an invertible matrix $S$ and a diagonal matrix $D$ such that […]
- Are Coefficient Matrices of the Systems of Linear Equations Nonsingular?
(a) Suppose that a $3\times 3$ system of linear equations is inconsistent. Is the coefficient matrix of the system nonsingular?
(b) Suppose that a $3\times 3$ homogeneous system of linear equations has a solution $x_1=0, x_2=-3, x_3=5$. Is the coefficient matrix of the system […]
- How to Calculate and Simplify a Matrix Polynomial
Let $T=\begin{bmatrix}
1 & 0 & 2 \\
0 &1 &1 \\
0 & 0 & 2
\end{bmatrix}$.
Calculate and simplify the expression
\[-T^3+4T^2+5T-2I,\]
where $I$ is the $3\times 3$ identity matrix.
(The Ohio State University Linear Algebra Exam)
Hint.
Use the […]