Determine the Dimension of a Mysterious Vector Space From Coordinate Vectors
Let $V$ be a vector space and $B$ be a basis for $V$.
Let $\mathbf{w}_1, \mathbf{w}_2, \mathbf{w}_3, \mathbf{w}_4, \mathbf{w}_5$ be vectors in $V$.
Suppose that $A$ is the matrix whose columns are the coordinate vectors of $\mathbf{w}_1, \mathbf{w}_2, \mathbf{w}_3, […]
True or False Quiz About a System of Linear Equations
(Purdue University Linear Algebra Exam)
Which of the following statements are true?
(a) A linear system of four equations in three unknowns is always inconsistent.
(b) A linear system with fewer equations than unknowns must have infinitely many solutions.
(c) […]
Every Group of Order 24 Has a Normal Subgroup of Order 4 or 8
Prove that every group of order $24$ has a normal subgroup of order $4$ or $8$.
Proof.
Let $G$ be a group of order $24$.
Note that $24=2^3\cdot 3$.
Let $P$ be a Sylow $2$-subgroup of $G$. Then $|P|=8$.
Consider the action of the group $G$ on […]
Matrix Representations for Linear Transformations of the Vector Space of Polynomials
Let $P_2(\R)$ be the vector space over $\R$ consisting of all polynomials with real coefficients of degree $2$ or less.
Let $B=\{1,x,x^2\}$ be a basis of the vector space $P_2(\R)$.
For each linear transformation $T:P_2(\R) \to P_2(\R)$ defined below, find the matrix representation […]
Every Prime Ideal of a Finite Commutative Ring is Maximal
Let $R$ be a finite commutative ring with identity $1$. Prove that every prime ideal of $R$ is a maximal ideal of $R$.
Proof.
We give two proofs. The first proof uses a result of a previous problem. The second proof is self-contained.
Proof 1.
Let $I$ be a prime ideal […]
A Matrix Similar to a Diagonalizable Matrix is Also Diagonalizable
Let $A, B$ be matrices. Show that if $A$ is diagonalizable and if $B$ is similar to $A$, then $B$ is diagonalizable.
Definitions/Hint.
Recall the relevant definitions.
Two matrices $A$ and $B$ are similar if there exists a nonsingular (invertible) matrix $S$ such […]
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 […]