Tagged: system of linear equations

Quiz 1. Gauss-Jordan Elimination / Homogeneous System. Math 2568 Spring 2017.

Problem 262

(a) Solve the following system by transforming the augmented matrix to reduced echelon form (Gauss-Jordan elimination). Indicate the elementary row operations you performed.
\begin{align*}
x_1+x_2-x_5&=1\\
x_2+2x_3+x_4+3x_5&=1\\
x_1-x_3+x_4+x_5&=0
\end{align*}

(b) Determine all possibilities for the solution set of a homogeneous system of $2$ equations in $2$ unknowns that has a solution $x_1=1, x_2=5$.

 
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Basis and Dimension of the Subspace of All Polynomials of Degree 4 or Less Satisfying Some Conditions.

Problem 256

Let $P_4$ be the vector space consisting of all polynomials of degree $4$ or less with real number coefficients.
Let $W$ be the subspace of $P_2$ by
\[W=\{ p(x)\in P_4 \mid p(1)+p(-1)=0 \text{ and } p(2)+p(-2)=0 \}.\] Find a basis of the subspace $W$ and determine the dimension of $W$.

 
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Find Values of $a$ so that Augmented Matrix Represents a Consistent System

Problem 249

Suppose that the following matrix $A$ is the augmented matrix for a system of linear equations.
\[A= \left[\begin{array}{rrr|r}
1 & 2 & 3 & 4 \\
2 &-1 & -2 & a^2 \\
-1 & -7 & -11 & a
\end{array} \right],\] where $a$ is a real number. Determine all the values of $a$ so that the corresponding system is consistent.

 
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A Matrix Representation of a Linear Transformation and Related Subspaces

Problem 164

Let $T:\R^4 \to \R^3$ be a linear transformation defined by
\[ T\left (\, \begin{bmatrix}
x_1 \\
x_2 \\
x_3 \\
x_4
\end{bmatrix} \,\right) = \begin{bmatrix}
x_1+2x_2+3x_3-x_4 \\
3x_1+5x_2+8x_3-2x_4 \\
x_1+x_2+2x_3
\end{bmatrix}.\]

(a) Find a matrix $A$ such that $T(\mathbf{x})=A\mathbf{x}$.

(b) Find a basis for the null space of $T$.

(c) Find the rank of the linear transformation $T$.

(The Ohio State University Linear Algebra Exam Problem)
 
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Possibilities For the Number of Solutions for a Linear System

Problem 102

Determine whether the following systems of equations (or matrix equations) described below has no solution, one unique solution or infinitely many solutions and justify your answer.


(a) \[\left\{
\begin{array}{c}
ax+by=c \\
dx+ey=f,
\end{array}
\right.
\] where $a,b,c, d$ are scalars satisfying $a/d=b/e=c/f$.


(b) $A \mathbf{x}=\mathbf{0}$, where $A$ is a singular matrix.


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


(d) $A\mathbf{x}=\mathbf{b}$, where the row-reduced echelon form of the augmented matrix $[A|\mathbf{b}]$ looks as follows:
\[\begin{bmatrix}
1 & 0 & -1 & 0 \\
0 &1 & 2 & 0 \\
0 & 0 & 0 & 1
\end{bmatrix}.\] (The Ohio State University, Linear Algebra Exam)
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The Vector Space Consisting of All Traceless Diagonal Matrices

Problem 79

Let $V$ be the set of all $n \times n$ diagonal matrices whose traces are zero.
That is,

\begin{equation*}
V:=\left\{ A=\begin{bmatrix}
a_{11} & 0 & \dots & 0 \\
0 &a_{22} & \dots & 0 \\
0 & 0 & \ddots & \vdots \\
0 & 0 & \dots & a_{nn}
\end{bmatrix} \quad \middle| \quad
\begin{array}{l}
a_{11}, \dots, a_{nn} \in \C,\\
\tr(A)=0 \\
\end{array}
\right\}
\end{equation*}

Let $E_{ij}$ denote the $n \times n$ matrix whose $(i,j)$-entry is $1$ and zero elsewhere.

(a) Show that $V$ is a subspace of the vector space $M_n$ over $\C$ of all $n\times n$ matrices. (You may assume without a proof that $M_n$ is a vector space.)

(b) Show that matrices
\[E_{11}-E_{22}, \, E_{22}-E_{33}, \, \dots,\, E_{n-1\, n-1}-E_{nn}\] are a basis for the vector space $V$.

(c) Find the dimension of $V$.
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True or False Quiz About a System of Linear Equations

Problem 78

Determine whether the following sentence is True or False.

(Purdue University Linear Algebra Exam)

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Conditions on Coefficients that a Matrix is Nonsingular

Problem 72

(a) Let $A=(a_{ij})$ be an $n\times n$ matrix. Suppose that the entries of the matrix $A$ satisfy the following relation.
\[|a_{ii}|>|a_{i1}|+\cdots +|a_{i\,i-1}|+|a_{i \, i+1}|+\cdots +|a_{in}|\] for all $1 \leq i \leq n$.
Show that the matrix $A$ is nonsingular.

(b) Let $B=(b_{ij})$ be an $n \times n$ matrix whose entries satisfy the relation
\[ |b_{i\,i}|=1 \hspace{0.5cm} \text{ and }\hspace{0.5cm} |b_{ij}|<\frac{1}{n-1}\] for all $i$ and $j$ with $i \neq j$.
Prove that the matrix $B$ is nonsingular.

(c)
Determine whether the following matrix is nonsingular or not.
\[C=\begin{bmatrix}
\pi & e & e^2/2\pi^2 \\[5 pt] e^2/2\pi^2 &\pi &e \\[5pt] e & e^2/2\pi^2 & \pi
\end{bmatrix},\] where $\pi=3.14159\dots$, and $e=2.71828\dots$ is Euler’s number (or Napier’s constant).

 

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Linear Independent Vectors, Invertible Matrix, and Expression of a Vector as a Linear Combinations

Problem 66

Consider the matrix
\[A=\begin{bmatrix}
1 & 2 & 1 \\
2 &5 &4 \\
1 & 1 & 0
\end{bmatrix}.\]


(a) Calculate the inverse matrix $A^{-1}$. If you think the matrix $A$ is not invertible, then explain why.


(b) Are the vectors
\[ \mathbf{A}_1=\begin{bmatrix}
1 \\
2 \\
1
\end{bmatrix}, \mathbf{A}_2=\begin{bmatrix}
2 \\
5 \\
1
\end{bmatrix},
\text{ and } \mathbf{A}_3=\begin{bmatrix}
1 \\
4 \\
0
\end{bmatrix}\] linearly independent?


(c) Write the vector $\mathbf{b}=\begin{bmatrix}
1 \\
1 \\
1
\end{bmatrix}$ as a linear combination of $\mathbf{A}_1$, $\mathbf{A}_2$, and $\mathbf{A}_3$.

(The Ohio State University, Linear Algebra Exam)

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Solving a System of Linear Equations By Using an Inverse Matrix

Problem 65

Consider the system of linear equations
\begin{align*}
x_1&= 2, \\
-2x_1 + x_2 &= 3, \\
5x_1-4x_2 +x_3 &= 2
\end{align*}

(a) Find the coefficient matrix and its inverse matrix.

(b) Using the inverse matrix, solve the system of linear equations.

(The Ohio State University, Linear Algebra Exam)

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Find All Matrices Satisfying a Given Relation

Problem 43

Let $a$ and $b$ be two distinct positive real numbers. Define matrices
\[A:=\begin{bmatrix}
0 & a\\
a & 0
\end{bmatrix}, \,\,
B:=\begin{bmatrix}
0 & b\\
b& 0
\end{bmatrix}.\]

Find all the pairs $(\lambda, X)$, where $\lambda$ is a real number and $X$ is a non-zero real matrix satisfying the relation
\[AX+XB=\lambda X. \tag{*} \]

 

(The University of Tokyo Linear Algebra Exam)

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