system-of-linear-equations-eye-catch
system-of-linear-equations-eye-catch
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- If $\mathbf{v}, \mathbf{w}$ are Linearly Independent Vectors and $A$ is Nonsingular, then $A\mathbf{v}, A\mathbf{w}$ are Linearly Independent
Let $A$ be an $n\times n$ nonsingular matrix. Let $\mathbf{v}, \mathbf{w}$ be linearly independent vectors in $\R^n$. Prove that the vectors $A\mathbf{v}$ and $A\mathbf{w}$ are linearly independent.
Proof.
Suppose that we have a linear […]
- 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, […]
- Equivalent Conditions For a Prime Ideal in a Commutative Ring
Let $R$ be a commutative ring and let $P$ be an ideal of $R$. Prove that the following statements are equivalent:
(a) The ideal $P$ is a prime ideal.
(b) For any two ideals $I$ and $J$, if $IJ \subset P$ then we have either $I \subset P$ or $J \subset P$.
Proof. […]
- Find a Matrix that Maps Given Vectors to Given Vectors
Suppose that a real matrix $A$ maps each of the following vectors
\[\mathbf{x}_1=\begin{bmatrix}
1 \\
1 \\
1
\end{bmatrix}, \mathbf{x}_2=\begin{bmatrix}
0 \\
1 \\
1
\end{bmatrix}, \mathbf{x}_3=\begin{bmatrix}
0 \\
0 \\
1
\end{bmatrix} \]
into the […]
- The Inverse Image of an Ideal by a Ring Homomorphism is an Ideal
Let $f:R\to R'$ be a ring homomorphism. Let $I'$ be an ideal of $R'$ and let $I=f^{-1}(I)$ be the preimage of $I$ by $f$. Prove that $I$ is an ideal of the ring $R$.
Proof.
To prove $I=f^{-1}(I')$ is an ideal of $R$, we need to check the following two […]
- Matrix Representation, Rank, and Nullity of a Linear Transformation $T:\R^2\to \R^3$
Let $T:\R^2 \to \R^3$ be a linear transformation such that
\[T\left(\, \begin{bmatrix}
3 \\
2
\end{bmatrix} \,\right)
=\begin{bmatrix}
1 \\
2 \\
3
\end{bmatrix} \text{ and }
T\left(\, \begin{bmatrix}
4\\
3
\end{bmatrix} […]
- Prove the Ring Isomorphism $R[x,y]/(x) \cong R[y]$
Let $R$ be a commutative ring. Consider the polynomial ring $R[x,y]$ in two variables $x, y$.
Let $(x)$ be the principal ideal of $R[x,y]$ generated by $x$.
Prove that $R[x, y]/(x)$ is isomorphic to $R[y]$ as a ring.
Proof.
Define the map $\psi: R[x,y] \to […]
- Lower and Upper Bounds of the Probability of the Intersection of Two Events
Let $A, B$ be events with probabilities $P(A)=2/5$, $P(B)=5/6$, respectively. Find the best lower and upper bound of the probability $P(A \cap B)$ of the intersection $A \cap B$. Namely, find real numbers $a, b$ such that
\[a \leq P(A \cap B) \leq b\]
and $P(A \cap B)$ could […]