Question

Find out whether the given vectors are dependent or independent; if they are dependent, find a linearly independent subset. In other words, find the dimension of the subspace spanned by the vectors, and a basis for it. Write each of the given vectors as a linear combination of the basis vectors. $$ (1,-2,3),(1,1,1),(-2,1,-4),(3,0,5) $$

Short answer

The vectors form a 2-dimensional subspace with a basis \(( 1, -2, 3 )\) and \(( 1, 1, 1 )\). The vector \(( -2, 1, -4 )\) is \(-2(1, -2, 3)\) and \((3, 0, 5 )\) is \((1, -2, 3) + 2(1, 1, 1)\).

Step-by-step solution

Set Up the Vector Matrix

Arrange the vectors as columns of a matrix to form a matrix A: \[ A = \begin{bmatrix} 1 & 1 & -2 & 3 \ -2 & 1 & 1 & 0 \ 3 & 1 & -4 & 5 \end{bmatrix} \]

Perform Row Reduction

Use Gaussian elimination to row reduce the matrix to its row echelon form. Perform the row operations on A: \[ \begin{bmatrix} 1 & 1 & -2 & 3 \ -2 & 1 & 1 & 0 \ 3 & 1 & -4 & 5 \end{bmatrix} \xrightarrow{R_2 + 2R_1} \begin{bmatrix} 1 & 1 & -2 & 3 \ 0 & 3 & -3 & 6 \ 3 & 1 & -4 & 5 \end{bmatrix} \xrightarrow{R_3 - 3R_1} \begin{bmatrix} 1 & 1 & -2 & 3 \ 0 & 3 & -3 & 6 \ 0 & -2 & 2 & -4 \end{bmatrix} \] Now simplify further: \[ \begin{bmatrix} 1 & 1 & -2 & 3 \ 0 & 3 & -3 & 6 \ 0 & -2 & 2 & -4 \end{bmatrix} \xrightarrow{R_3 + \frac{2}{3}R_2} \begin{bmatrix} 1 & 1 & -2 & 3 \ 0 & 3 & -3 & 6 \ 0 & 0 & 0 & 0 \end{bmatrix} \]

Determine Linear Independence

The row-reduced form of the matrix shows two non-zero rows, indicating two pivot positions. This means there are 2 linearly independent vectors, and the vectors span a subspace of dimension 2.

Identify a Basis for the Subspace

The first and second columns correspond to the pivot positions. Hence, the vectors \((1, -2, 3)\) and \((1, 1, 1)\) form a basis for the subspace.

Express Other Vectors as Linear Combinations

To express \(( -2, 1, -4 )\) and \((3, 0, 5)\) as linear combinations of the basis vectors, solve the system: For \(( -2, 1, -4 )\): \[-2 = a(1) + b(1)\] \[1 = a(-2) + b(1)\] \[-4 = a(3) + b(1)\] Solving gives: \[ a = -2, b = 0 \] So, \(( -2, 1, -4 )\) can be written as \[-2(1, -2, 3)\]. For \((3, 0, 5)\): \[3 = a(1) + b(1)\] \[0 = a(-2) + b(1)\] \[5 = a(3) + b(1)\] Solving gives: \[ a = 1, b = 2 \] So, \((3, 0, 5 )\) can be written as \[(1, -2, 3) + 2(1, 1, 1)\].

Key concepts

Vector Dependence

Understanding vector dependence is important in linear algebra. Vectors are **dependent** if at least one of them can be expressed as a linear combination of the others. If none of the vectors can be written this way, they are **independent**.
To check for dependence or independence among vectors, you usually arrange them as columns in a matrix and use matrix operations to determine the number of pivot positions.
For example, if you have a set of vectors and find that one can be written as a combination of others, they are dependent. Knowing whether vectors are dependent or independent helps determine the structure and dimensions of spaces they span.

Gaussian Elimination

Gaussian elimination is a method used to solve systems of linear equations. It involves performing row operations to transform a matrix into row echelon form or reduced row echelon form.
This method helps identify pivot positions in a matrix, indicating the number of linearly independent vectors in the set.
When you reduce a matrix, you'll perform several steps:

• Swapping rows
• Scaling rows by non-zero constants
• Adding or subtracting multiples of rows

Each of these operations is aimed at simplifying the matrix. The goal is to have ones (pivots) on the diagonal and zeros below them.

Basis Vectors

A **basis** of a vector space is a set of vectors that are linearly independent and span the vector space. To form a basis, the vectors must meet two criteria:

• **Linear independence**: No vector in the set can be written as a combination of the others
• **Span**: The set of vectors can represent any vector in the space through linear combinations

In our example, the vectors \( (1, -2, 3) \) and \( (1, 1, 1) \) form a basis because they are the pivot columns after performing Gaussian elimination. They span a subspace, meaning any vector in that space can be written as a combination of these basis vectors.

Dimension of Subspace

The **dimension** of a subspace is the number of vectors in its basis. It tells how many coordinates are needed to specify any vector in that subspace.
If a subspace is spanned by two vectors, its dimension is 2. In our example, after row reducing the matrix, we find two pivot positions. Therefore, the subspace spanned by the given vectors is of dimension 2.
This dimension tells us that any vector in this subspace can be represented by a combination of two basis vectors. Understanding dimensions helps comprehend the complexity and size of the space in which the vectors lie.