https://r2.erweima.ai/mathbot/image/1727007755688p89haw0g.jpg

Let's go through each part of the question from the image you uploaded. For each part (a)–(h), I'll determine whether the statement is true or false, and provide justifications.

### (a) A linear system whose equations are all homogeneous must be consistent.
**True**  
A homogeneous system has the form \( Ax = 0 \), where \( A \) is a matrix and \( 0 \) is the zero vector. The trivial solution (where all variables are zero) always satisfies this system, meaning it is always consistent. 

### (b) Multiplying a row of an augmented matrix by zero is an acceptable elementary row operation.
**False**  
Multiplying a row by zero eliminates all the information from that row, which is not a valid elementary row operation. Elementary operations include multiplying a row by a non-zero scalar, swapping rows, and adding a multiple of one row to another.

### (c) The linear system:
\[
x - y = 3
\]
\[
2x - 2y = k
\]
cannot have a unique solution, regardless of the value of \( k \).
**True**  
The second equation is a scalar multiple of the first. Therefore, the system represents the same line, and the two equations are dependent. They either have infinitely many solutions (if \( k = 6 \)) or no solution (if \( k \neq 6 \)), but never a unique solution.

### (d) A single linear equation with two or more unknowns must have infinitely many solutions.
**True**  
A single equation with more than one unknown typically represents a line (in two dimensions) or a plane (in three dimensions), meaning there are infinitely many points (solutions) that satisfy the equation.

### (e) If the number of equations in a linear system exceeds the number of unknowns, then the system must be inconsistent.
**False**  
The system could still be consistent, depending on the equations. For example, if some of the equations are redundant (linear combinations of others), the system can still have a solution, even if there are more equations than unknowns.

### (f) If each equation in a consistent linear system is multiplied by a constant \( c \), then all solutions to the new system can be obtained by multiplying solutions from the original system by \( c \).
**False**  
Multiplying each equation by a constant \( c \) does not change the solution set of the system. The solutions remain the same; they are not scaled by \( c \).

### (g) Elementary row operations permit one row of an augmented matrix to be subtracted from another.
**True**  
One of the elementary row operations is adding or subtracting a multiple of one row to or from another. Therefore, subtracting one row from another is valid.

### (h) The linear system with the corresponding augmented matrix:
\[
\begin{bmatrix} 
2 & -1 & 4 \\
0 & 0 & -1 
\end{bmatrix}
\]
has infinitely many solutions.
**False**  
The second row implies \( 0 = -1 \), which is a contradiction. This means the system is inconsistent and has no solutions.

---

Would you like further details on any of these answers?

### Here are 5 related questions to expand your understanding:
1. What is the definition of a consistent and inconsistent linear system?
2. Can you give an example of a homogeneous system with non-trivial solutions?
3. What are the acceptable elementary row operations and their importance?
4. Why does the system in part (c) not have a unique solution, and how can you verify it algebraically?
5. What does it mean geometrically when a system of linear equations has infinitely many solutions?

### Tip:  
In solving linear systems, always check for row inconsistencies (like zero rows leading to contradictions) to quickly identify if the system is inconsistent.

In parts (a)–(h), determine whether the statement is true or false, and justify your answer. (a) A linear system whose equations are all homogeneous must be consistent. (b) Multiplying a row of an augmented matrix through by zero is an acceptable elementary row operation. (c) The linear system (x - y = 3, 2x - 2y = k) cannot have a unique solution, regardless of the value of k. (d) A single linear equation with two or more unknowns must have infinitely many solutions. (e) If the number of equations in a linear system exceeds the number of unknowns, then the system must be inconsistent. (f) If each equation in a consistent linear system is multiplied by a constant c, then all solutions to the new system can be obtained by multiplying solutions from the original system by c. (g) Elementary row operations permit one row of an augmented matrix to be subtracted from another. (h) The linear system with corresponding augmented matrix [[2, -1, 4], [0, 0, -1]] has infinitely many solutions.

Determine the truth of statements related to linear systems, elementary row operations, and solutions to systems of equations. Concepts covered include consistency of homogeneous systems, augmented matrices, and row operations. Ideal for students studying Linear Algebra at the undergraduate level.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation