# Purdue-Algebra-Exam-eye-catch

by Yu ·

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- The Rank and Nullity of a Linear Transformation from Vector Spaces of Matrices to Polynomials Let $V$ be the vector space of $2 \times 2$ matrices with real entries, and $\mathrm{P}_3$ the vector space of real polynomials of degree 3 or less. Define the linear transformation $T : V \rightarrow \mathrm{P}_3$ by \[T \left( \begin{bmatrix} a & b \\ c & d \end{bmatrix} \right) = […]
- Show that the Given 2 by 2 Matrix is Singular Consider the matrix $M = \begin{bmatrix} 1 & 4 \\ 3 & 12 \end{bmatrix}$. (a) Show that $M$ is singular. (b) Find a non-zero vector $\mathbf{v}$ such that $M \mathbf{v} = \mathbf{0}$, where $\mathbf{0}$ is the $2$-dimensional zero vector. Solution. (a) Show […]
- All the Conjugacy Classes of the Dihedral Group $D_8$ of Order 8 Determine all the conjugacy classes of the dihedral group \[D_{8}=\langle r,s \mid r^4=s^2=1, sr=r^{-1}s\rangle\] of order $8$. Hint. You may directly compute the conjugates of each element but we are going to use the following theorem to simplify the […]
- Prove that $(A + B) \mathbf{v} = A\mathbf{v} + B\mathbf{v}$ Using the Matrix Components Let $A$ and $B$ be $n \times n$ matrices, and $\mathbf{v}$ an $n \times 1$ column vector. Use the matrix components to prove that $(A + B) \mathbf{v} = A\mathbf{v} + B\mathbf{v}$. Solution. We will use the matrix components $A = (a_{i j})_{1 \leq i, j \leq n}$, $B = […]
- Is the Map $T(f)(x) = (f(x))^2$ a Linear Transformation from the Vector Space of Real Functions? Let $C (\mathbb{R})$ be the vector space of real functions. Define the map $T$ by $T(f)(x) = (f(x))^2$ for $f \in C(\mathbb{R})$. Determine if $T$ is a linear transformation or not. If it is, determine the range of $T$. Solution. We claim that $T$ is not a […]
- Is an Eigenvector of a Matrix an Eigenvector of its Inverse? Suppose that $A$ is an $n \times n$ matrix with eigenvalue $\lambda$ and corresponding eigenvector $\mathbf{v}$. (a) If $A$ is invertible, is $\mathbf{v}$ an eigenvector of $A^{-1}$? If so, what is the corresponding eigenvalue? If not, explain why not. (b) Is $3\mathbf{v}$ an […]
- Dimension of the Sum of Two Subspaces Let $U$ and $V$ be finite dimensional subspaces in a vector space over a scalar field $K$. Then prove that \[\dim(U+V) \leq \dim(U)+\dim(V).\] Definition (The sum of subspaces). Recall that the sum of subspaces $U$ and $V$ is \[U+V=\{\mathbf{x}+\mathbf{y} \mid […]
- Quiz 11. Find Eigenvalues and Eigenvectors/ Properties of Determinants (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 […]