Linear Algebra: Systems of Equations

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Questions and Answers

What defines a consistent system of equations?

It has multiple solutions but not infinite.

It has only one unique solution.

It has at least one solution. (correct)

It has no solutions.

In which case does a system of linear equations have a unique solution?

When the equations intersect at two points.

When the planes defined by the equations are parallel.

When the equations intersect at exactly one point. (correct)

When the equations represent parallel lines.

Which of the following does NOT represent a system of linear equations?

$3x + 4y = 12$

$y = 3x^2$ (correct)

$2x - y = 1$

$x + 2y = 4$

What is the graphical representation of a system of equations with no solutions?

Two parallel lines that never meet. Signup and view all the answers

What is the significance of row echelon form in solving a system of linear equations?

It simplifies the matrix for easier computation. Signup and view all the answers

Which operation is NOT valid while performing row reduction on a matrix?

Dividing a row by zero. Signup and view all the answers

How are infinite solutions represented in a system of linear equations?

By identical lines that overlap completely. Signup and view all the answers

What best describes an inconsistent system of equations?

It has no common solution points. Signup and view all the answers

What can be concluded if the vector w is not in the span of vectors v1 and v2?

The equation Ax = b has no solutions. Signup and view all the answers

What is true about two noncollinear vectors v and w?

They are linearly independent. Signup and view all the answers

When can we ignore the right-hand side when solving the equation Ax = 0?

When it is a homogeneous equation. Signup and view all the answers

What happens to the span when a linearly independent vector is added to a set of vectors?

The span increases. Signup and view all the answers

For a set of vectors to be considered linearly dependent, which condition must hold?

At least one vector can be expressed as a linear combination of the others. Signup and view all the answers

In which scenario will the homogeneous equation Ax = b have a nontrivial solution?

When the matrix A is singular. Signup and view all the answers

If vectors v and w are in span {u, v}, which of the following can be inferred about their linear independence?

Vectors v and w are dependent on each other. Signup and view all the answers

What is the consequence of having three vectors {v, w, u} that are linearly dependent?

At least one vector can be expressed as a linear combination of the other two. Signup and view all the answers

Study Notes

Linear Algebra

Linear refers to lines, planes, etc.
Algebra refers to solving equations with unknowns.
A system of linear equations is a set of equations with the same variables, where each equation represents a line or plane in a multi-dimensional space.
A system of linear equations can be solved both algebraically (by manipulating equations) and geometrically (by visualizing the intersection of lines or planes).
Examples of linear equations:
- x + y = 5
- 2x - 3y = 1
- 4x + 2y - z = 7
Examples of non-linear equations:
- x² + y² = 9
- y = sin(x)
A system of linear equations can have:
- One solution (unique solution): The lines or planes intersect at a single point.
- No solution (inconsistent system): The lines or planes are parallel and do not intersect.
- Infinitely many solutions: The lines or planes coincide (overlap).
The elimination method is a technique to solve systems of linear equations by eliminating variables through algebraic manipulations.
Row reduction is a systematic process used to transform a system of linear equations into an equivalent system that is easier to solve.
A matrix is a rectangular array of numbers arranged in rows and columns.
An augmented matrix is a matrix that represents a system of linear equations, where the coefficients of the variables and the constants are arranged in a specific way.
A matrix is in row echelon form when:
- All rows that contain only zeros are at the bottom.
- The first nonzero element (pivot) in each nonzero row is 1.
- The pivots are in a staircase pattern.
Reduced row echelon form: A more simplified form of row echelon form where the pivot is the only nonzero element in its column.
When a non-augmented column in a reduced row echelon form lacks a pivot, it indicates the system has multiple solutions.
Span: The span of a set of vectors is the set of all possible linear combinations of those vectors.
A vector is in the span of a set of vectors if it can be expressed as a linear combination of them.
Linearly independent vectors: A set of vectors is linearly independent if no vector in the set can be expressed as a linear combination of the others.
Linearly dependent vectors: If at least one vector in a set can be expressed as a linear combination of the others, the set is linearly dependent.
Homogeneous equation: An equation where the constant term is zero (e.g., Ax = 0).
The solution set of a homogeneous equation always includes the trivial solution (x = 0).
The solution set of a non-homogeneous equation (Ax = b) is a translate of the solution set of the corresponding homogeneous equation (Ax = 0) by the particular solution 'p' (Ax = p).

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Description

This quiz focuses on systems of linear equations, exploring both algebraic and geometric solutions. Learn to differentiate between unique solutions, inconsistent systems, and cases with infinitely many solutions, through examples and methods like elimination.