Problem 9 - Differential Equations and Linear Algebra

Question

determine whether the given set of vectors is linearly independent or linearly dependent in \(\mathbb{R}^{n} .\) In the case of linear dependence, find a dependency relationship. $$\\{(1,-1,2,3),(2,-1,1,-1),(-1,1,1,1)\\}$$.

Step-by-Step Solution

Answer

The given set of vectors is linearly independent in \(\mathbb{R}^{n}\), as the row echelon form of the augmented matrix has no free variables.

Step 1: Form the augmented matrix

To check for independence, we will form an augmented matrix from the given set of vectors and perform Gaussian elimination: $$ \begin{bmatrix} 1 & 2 & -1 \\ -1 & -1 & 1 \\ 2 & 1 & 1 \\ 3 & -1 & 1 \\ \end{bmatrix} $$

Step 2: Perform Gaussian elimination

Now let's perform row operations to obtain row echelon form: Swap first and second row: $$ \begin{bmatrix} -1 & -1 & 1 \\ 1 & 2 & -1 \\ 2 & 1 & 1 \\ 3 & -1 & 1 \\ \end{bmatrix} $$ Add first row multiplied by 1 to the second row, and multiply second row with -1: $$ \begin{bmatrix} -1 & -1 & 1 \\ 0 & 3 & -2 \\ 2 & 1 & 1 \\ 3 & -1 & 1 \\ \end{bmatrix} $$ Add first row to the fourth row, and add the first row multiplied by -2 to the third row: $$ \begin{bmatrix} -1 & -1 & 1 \\ 0 & 3 & -2 \\ 0 & 3 & -1 \\ 0 & -2 & 2 \\ \end{bmatrix} $$ Swap second and third row, and then swap third and fourth one: $$ \begin{bmatrix} -1 & -1 & 1 \\ 0 & 3 & -1 \\ 0 & -2 & 2 \\ 0 & 3 & -2 \\ \end{bmatrix} $$ Add third row multiplied by -1 and divided by 2 to the second row, and then add this new third row multiplied by -3 to the fourth one: $$ \begin{bmatrix} -1 & -1 & 1 \\ 0 & 3 & -1 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \\ \end{bmatrix} $$

Step 3: Analyze the row echelon form

With the row echelon form, we can see that there are no free variables (i.e., all columns have pivots). This means that the set of vectors is linearly independent, and there is no dependency relationship.

Key Concepts

Gaussian elimination

Gaussian elimination is a systematic method for solving systems of linear equations, but it's also a powerful tool for determining the linear independence of vectors. This procedure transforms a matrix into row echelon form by performing row operations that include swapping rows, multiplying a row by a non-zero scalar, and adding a scalar multiple of one row to another row.

The goal of Gaussian elimination is to simplify the matrix to a point where the solution to the system of equations (if one exists) becomes obvious or to reveal the system has no solution or an infinite number of solutions. With vectors, we can employ Gaussian elimination on the matrix formed by placing the vectors as columns to find out if the vectors are linearly independent or dependent.

Row echelon form

A matrix is in row echelon form when it satisfies the following conditions: each non-zero row has a leading term (also known as a pivot) which is to the right of the leading term in the row above it, and any rows consisting entirely of zeros are at the bottom of the matrix. In a row echelon form, the leading term of each row is also typically the only non-zero entry in its column.

Significance in Linear Independence

In the context of linear independence, achieving row echelon form allows you to determine if any of the vectors (columns in the original matrix) can be expressed as linear combinations of the others. If there is a pivot in every original column of the matrix, this indicates that the vectors are linearly independent.

Dependency relationship

A dependency relationship among vectors implies that at least one of the vectors can be written as a combination of the others. This is a sign of linear dependence. If we have a set of vectors, and we can find coefficients (not all zero) such that a linear combination of these vectors equals the zero vector, then they are linearly dependent.

The process of Gaussian elimination can reveal any dependency relationships. If after applying row operations, there is a row of zeros corresponding to a vector, it indicates that there is at least one dependency relationship among the vectors. The coefficients that lead to this zero row are the coefficients of the dependency relationship.

Augmented matrix

An augmented matrix is commonly used in the context of solving systems of linear equations, where it includes the coefficients of the variables and the constant terms from the equations, separated by a line. However, to assess linear independence, the augmented matrix consists solely of the vectors in question.

In this linear independence exercise, we discarded the traditional augmented matrix's 'constants column' as the main goal is to only assess the relationship between vectors. Thus, the matrix is augmented only in the sense that the vectors are combined into one matrix – and not augmented by additional columns. This matrix is then subjected to Gaussian elimination to determine the vectors' independence or dependence.