# vector-space

by Yu · Published · Updated

Add to solve later

Add to solve later

Add to solve later

### More from my site

- If a Group is of Odd Order, then Any Nonidentity Element is Not Conjugate to its Inverse Let $G$ be a finite group of odd order. Assume that $x \in G$ is not the identity element. Show that $x$ is not conjugate to $x^{-1}$. Proof. Assume the contrary, that is, assume that there exists $g \in G$ such that $gx^{-1}g^{-1}=x$. Then we have \[xg=gx^{-1}. […]
- A Linear Transformation is Injective (One-To-One) if and only if the Nullity is Zero Let $U$ and $V$ be vector spaces over a scalar field $\F$. Let $T: U \to V$ be a linear transformation. Prove that $T$ is injective (one-to-one) if and only if the nullity of $T$ is zero. Definition (Injective, One-to-One Linear Transformation). A linear […]
- Find an Orthonormal Basis of the Range of a Linear Transformation Let $T:\R^2 \to \R^3$ be a linear transformation given by \[T\left(\, \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \,\right) = \begin{bmatrix} x_1-x_2 \\ x_2 \\ x_1+ x_2 \end{bmatrix}.\] Find an orthonormal basis of the range of $T$. (The Ohio […]
- Linear Combination of Eigenvectors is Not an Eigenvector Suppose that $\lambda$ and $\mu$ are two distinct eigenvalues of a square matrix $A$ and let $\mathbf{x}$ and $\mathbf{y}$ be eigenvectors corresponding to $\lambda$ and $\mu$, respectively. If $a$ and $b$ are nonzero numbers, then prove that $a \mathbf{x}+b\mathbf{y}$ is not an […]
- Prove a Given Subset is a Subspace and Find a Basis and Dimension Let \[A=\begin{bmatrix} 4 & 1\\ 3& 2 \end{bmatrix}\] and consider the following subset $V$ of the 2-dimensional vector space $\R^2$. \[V=\{\mathbf{x}\in \R^2 \mid A\mathbf{x}=5\mathbf{x}\}.\] (a) Prove that the subset $V$ is a subspace of $\R^2$. (b) Find a basis for […]
- Example of a Nilpotent Matrix $A$ such that $A^2\neq O$ but $A^3=O$. Find a nonzero $3\times 3$ matrix $A$ such that $A^2\neq O$ and $A^3=O$, where $O$ is the $3\times 3$ zero matrix. (Such a matrix is an example of a nilpotent matrix. See the comment after the solution.) Solution. For example, let $A$ be the following $3\times […]
- If Two Matrices are Similar, then their Determinants are the Same Prove that if $A$ and $B$ are similar matrices, then their determinants are the same. Proof. Suppose that $A$ and $B$ are similar. Then there exists a nonsingular matrix $S$ such that \[S^{-1}AS=B\] by definition. Then we […]
- Stochastic Matrix (Markov Matrix) and its Eigenvalues and Eigenvectors (a) Let \[A=\begin{bmatrix} a_{11} & a_{12}\\ a_{21}& a_{22} \end{bmatrix}\] be a matrix such that $a_{11}+a_{12}=1$ and $a_{21}+a_{22}=1$. Namely, the sum of the entries in each row is $1$. (Such a matrix is called (right) stochastic matrix (also termed […]