Quiz 5: Example and Non-Example of Subspaces in 3-Dimensional Space

Problem 304

Problem 1 Let $W$ be the subset of the $3$-dimensional vector space $\R^3$ defined by
\[W=\left\{ \mathbf{x}=\begin{bmatrix}
x_1 \\
x_2 \\
x_3
\end{bmatrix}\in \R^3 \quad \middle| \quad 2x_1x_2=x_3 \right\}.\]

(a) Which of the following vectors are in the subset $W$? Choose all vectors that belong to $W$.
\[(1) \begin{bmatrix}
0 \\
0 \\
0
\end{bmatrix} \qquad(2) \begin{bmatrix}
1 \\
2 \\
2
\end{bmatrix} \qquad(3)\begin{bmatrix}
3 \\
0 \\
0
\end{bmatrix} \qquad(4) \begin{bmatrix}
0 \\
0
\end{bmatrix} \qquad(5) \begin{bmatrix}
1 & 2 & 4 \\
1 &2 &4
\end{bmatrix} \qquad(6) \begin{bmatrix}
1 \\
-1 \\
-2
\end{bmatrix}.\]

(b) Determine whether $W$ is a subspace of $\R^3$ or not.
 


Problem 2 Let $W$ be the subset of $\R^3$ defined by
\[W=\left\{ \mathbf{x}=\begin{bmatrix}
x_1 \\
x_2 \\
x_3
\end{bmatrix} \in \R^3 \quad \middle| \quad x_1=3x_2 \text{ and } x_3=0 \right\}.\] Determine whether the subset $W$ is a subspace of $\R^3$ or not.

 
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Given a Spanning Set of the Null Space of a Matrix, Find the Rank

Problem 303

Let $A$ be a real $7\times 3$ matrix such that its null space is spanned by the vectors
\[\begin{bmatrix}
1 \\
2 \\
0
\end{bmatrix}, \begin{bmatrix}
2 \\
1 \\
0
\end{bmatrix}, \text{ and } \begin{bmatrix}
1 \\
-1 \\
0
\end{bmatrix}.\] Then find the rank of the matrix $A$.

(Purdue University, Linear Algebra Final Exam Problem)
 
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Generators of the Augmentation Ideal in a Group Ring

Problem 302

Let $R$ be a commutative ring with $1$ and let $G$ be a finite group with identity element $e$. Let $RG$ be the group ring. Then the map $\epsilon: RG \to R$ defined by
\[\epsilon(\sum_{i=1}^na_i g_i)=\sum_{i=1}^na_i,\] where $a_i\in R$ and $G=\{g_i\}_{i=1}^n$, is a ring homomorphism, called the augmentation map and the kernel of $\epsilon$ is called the augmentation ideal.

(a) Prove that the augmentation ideal in the group ring $RG$ is generated by $\{g-e \mid g\in G\}$.

(b) Prove that if $G=\langle g\rangle$ is a finite cyclic group generated by $g$, then the augmentation ideal is generated by $g-e$.
 
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Solve a System by the Inverse Matrix and Compute $A^{2017}\mathbf{x}$

Problem 300

Let $A$ be the coefficient matrix of the system of linear equations
\begin{align*}
-x_1-2x_2&=1\\
2x_1+3x_2&=-1.
\end{align*}

(a) Solve the system by finding the inverse matrix $A^{-1}$.

(b) Let $\mathbf{x}=\begin{bmatrix}
x_1 \\
x_2
\end{bmatrix}$ be the solution of the system obtained in part (a).
Calculate and simplify
\[A^{2017}\mathbf{x}.\]

(The Ohio State University, Linear Algebra Midterm Exam Problem)
 
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Express a Vector as a Linear Combination of Given Three Vectors

Problem 298

Let
\[\mathbf{v}_1=\begin{bmatrix}
1 \\
5 \\
-1
\end{bmatrix}, \mathbf{v}_2=\begin{bmatrix}
1 \\
4 \\
3
\end{bmatrix}, \mathbf{v}_3=\begin{bmatrix}
1 \\
2 \\
1
\end{bmatrix}, \mathbf{b}=\begin{bmatrix}
2 \\
13 \\
6
\end{bmatrix}.\] Express the vector $\mathbf{b}$ as a linear combination of the vector $\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3$.

(The Ohio State University, Linear Algebra Midterm Exam Problem)
 
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Compute and Simplify the Matrix Expression Including Transpose and Inverse Matrices

Problem 297

Let $A, B, C$ be the following $3\times 3$ matrices.
\[A=\begin{bmatrix}
1 & 2 & 3 \\
4 &5 &6 \\
7 & 8 & 9
\end{bmatrix}, B=\begin{bmatrix}
1 & 0 & 1 \\
0 &3 &0 \\
1 & 0 & 5
\end{bmatrix}, C=\begin{bmatrix}
-1 & 0\ & 1 \\
0 &5 &6 \\
3 & 0 & 1
\end{bmatrix}.\] Then compute and simplify the following expression.
\[(A^{\trans}-B)^{\trans}+C(B^{-1}C)^{-1}.\]

(The Ohio State University, Linear Algebra Midterm Exam Problem)
 
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Solve the System of Linear Equations and Give the Vector Form for the General Solution

Problem 296

Solve the following system of linear equations and give the vector form for the general solution.
\begin{align*}
x_1 -x_3 -2x_5&=1 \\
x_2+3x_3-x_5 &=2 \\
2x_1 -2x_3 +x_4 -3x_5 &= 0
\end{align*}

(The Ohio State University, linear algebra midterm exam problem)
 
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The Possibilities For the Number of Solutions of Systems of Linear Equations that Have More Equations than Unknowns

Problem 295

Determine all possibilities for the number of solutions of each of the system of linear equations described below.

(a) A system of $5$ equations in $3$ unknowns and it has $x_1=0, x_2=-3, x_3=1$ as a solution.

(b) A homogeneous system of $5$ equations in $4$ unknowns and the rank of the system is $4$.
 

(The Ohio State University, Linear Algebra Midterm Exam Problem)
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Quiz 4: Inverse Matrix/ Nonsingular Matrix Satisfying a Relation

Problem 289

(a) Find the inverse matrix of
\[A=\begin{bmatrix}
1 & 0 & 1 \\
1 &0 &0 \\
2 & 1 & 1
\end{bmatrix}\] if it exists. If you think there is no inverse matrix of $A$, then give a reason.

(b) Find a nonsingular $2\times 2$ matrix $A$ such that
\[A^3=A^2B-3A^2,\] where
\[B=\begin{bmatrix}
4 & 1\\
2& 6
\end{bmatrix}.\] Verify that the matrix $A$ you obtained is actually a nonsingular matrix.

(The Ohio State University, Linear Algebra Midterm Exam Problem)
 
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Summary: Possibilities for the Solution Set of a System of Linear Equations

Problem 288

In this post, we summarize theorems about the possibilities for the solution set of a system of linear equations and solve the following problems.

Determine all possibilities for the solution set of the system of linear equations described below.

(a) A homogeneous system of $3$ equations in $5$ unknowns.

(b) A homogeneous system of $5$ equations in $4$ unknowns.

(c) A system of $5$ equations in $4$ unknowns.

(d) A system of $2$ equations in $3$ unknowns that has $x_1=1, x_2=-5, x_3=0$ as a solution.

(e) A homogeneous system of $4$ equations in $4$ unknowns.

(f) A homogeneous system of $3$ equations in $4$ unknowns.

(g) A homogeneous system that has $x_1=3, x_2=-2, x_3=1$ as a solution.

(h) A homogeneous system of $5$ equations in $3$ unknowns and the rank of the system is $3$.

(i) A system of $3$ equations in $2$ unknowns and the rank of the system is $2$.

(j) A homogeneous system of $4$ equations in $3$ unknowns and the rank of the system is $2$.
 
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Basis For Subspace Consisting of Matrices Commute With a Given Diagonal Matrix

Problem 287

Let $V$ be the vector space of all $3\times 3$ real matrices.
Let $A$ be the matrix given below and we define
\[W=\{M\in V \mid AM=MA\}.\] That is, $W$ consists of matrices that commute with $A$.
Then $W$ is a subspace of $V$.

Determine which matrices are in the subspace $W$ and find the dimension of $W$.

(a) \[A=\begin{bmatrix}
a & 0 & 0 \\
0 &b &0 \\
0 & 0 & c
\end{bmatrix},\] where $a, b, c$ are distinct real numbers.

(b) \[A=\begin{bmatrix}
a & 0 & 0 \\
0 &a &0 \\
0 & 0 & b
\end{bmatrix},\] where $a, b$ are distinct real numbers.

 
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