Question

Are the rows linearly independent in each of the following? \((a)\left[\begin{array}{rr}24 & 8 \\ 9 & -3\end{array}\right]\) (b) \(\left[\begin{array}{ll}2 & 0 \\ 0 & 2\end{array}\right]\) \((c)\left[\begin{array}{ll}0 & 4 \\ 3 & 2\end{array}\right]\) \((d)\left[\begin{array}{rr}-1 & 5 \\ 2 & -10\end{array}\right]\)

Short answer

(a), (b), and (c) have linearly independent rows; (d) does not.

Step-by-step solution

Determine Linear Independence Criterion

To determine if the rows of a matrix are linearly independent, we check if any row can be expressed as a linear combination of the others. For a 2x2 matrix, we simply check if the rows are scalar multiples of each other. If they are not, the rows are linearly independent.

Step 2a: Check Matrix (a)

Given matrix: \(\left[\begin{array}{rr}24 & 8 \ 9 & -3\end{array}\right]\). Check if \([9, -3]\) is a scalar multiple of \([24, 8]\). The condition is \(k[24, 8] = [9, -3]\) for some scalar \(k\). Solving gives \(k = \frac{9}{24} = \frac{3}{8}\) and \(-3 = \frac{3}{8} \times 8 = 3\), which does not hold. The rows are not scalar multiples; hence, they are independent.

Step 2b: Check Matrix (b)

Given matrix: \(\left[\begin{array}{ll}2 & 0 \ 0 & 2\end{array}\right]\). Each row has a non-zero element in a different column, so neither row can be a scalar multiple of the other. Therefore, the rows are linearly independent.

Step 2c: Check Matrix (c)

Given matrix: \(\left[\begin{array}{ll}0 & 4 \ 3 & 2\end{array}\right]\). Check if \([3, 2]\) is a scalar multiple of \([0, 4]\). The zero entry in one row must correspond to a zero entry in the multiple; since this is not possible, the rows are independent.

Step 2d: Check Matrix (d)

Given matrix: \(\left[\begin{array}{rr}-1 & 5 \ 2 & -10\end{array}\right]\). Check if \([2, -10]\) is a scalar multiple of \([-1, 5]\). Solving gives \(k=-2\) such that \(2 = -2(-1)\) and \(-10 = -2(5)\). Since both conditions hold, the rows are scalar multiples and thus linearly dependent.

Key concepts

Matrix Algebra

Matrix Algebra is a branch of mathematics that deals with matrices and the operations that can be performed on them. A matrix is essentially a rectangular array of numbers arranged in rows and columns. These arrays serve as a powerful tool in various fields, particularly in describing linear transformations and systems of linear equations.

Key operations in matrix algebra include addition, subtraction, and multiplication. Addition and subtraction are performed element-wise, meaning each corresponding entry is added or subtracted. Matrix multiplication, however, is more complex and involves the dot product of rows and columns.

When dealing with matrices, the concept of linear independence often arises, especially when analyzing whether the rows or columns of a matrix are related as scalar multiples of one another. Determining linear independence is crucial in various applications, such as understanding vector spaces in linear algebra.

Linear Algebra

Linear Algebra is the study of vectors, vector spaces, and linear mappings between these spaces. It's fundamental in mathematics because of its applications across numerous disciplines like physics, computer science, and engineering.

One core aspect of linear algebra is linear independence, a concept that determines whether a set of vectors (or rows in a matrix) cannot be expressed as a linear combination of other vectors. Understanding linear independence is pivotal as it indicates whether a set of vectors spans a space—meaning, whether you can form the entire vector space using these vectors.

In practical terms, a set of vectors is linearly independent if no vector can be written as a sum of scalar multiples of the others. This property, especially in matrix algebra, helps in solving system equations by identifying unique solutions.

Scalar Multiples

Scalar Multiples refer to the multiplication of a vector by a single number, known as a scalar. Given a vector, \\(\mathbf{v} = [v_1, v_2, ... , v_n]\), multiplying by a scalar \\(k\) produces a new vector \\(k\mathbf{v} = [kv_1, kv_2, ..., kv_n]\).

This process stretches or shrinks the original vector without altering its direction unless the scalar is negative, which would also reverse it. In the context of matrices, determining whether one row is a scalar multiple of another is a way to check for linear dependence.

If two rows in a matrix are scalar multiples, it means one can be obtained by multiplying the other by a constant. This relationship is a straightforward criterion used to evaluate linear independence. If no scalar can make one row equal to another, they are independent. Otherwise, the presence of a scalar multiple indicates dependency.

Row Vectors

Row Vectors are horizontal arrays of numbers within a matrix, representing one form of vector representation along with column vectors. Each row vector corresponds to an equation in a system of linear equations when using matrices to represent systems.

The examination of row vectors is crucial when determining linear independence in matrix algebra. By checking if any row vector is a scalar multiple of another, you ascertain whether the vectors are independent or dependent.

In settings like the given exercise, analyzing row vectors involves simple arithmetic to verify if one row vector can be expressed as a multiple of another. This simple check has deep implications, especially when identifying whether solutions to matrix equations are unique, leading to the representation of vectors and transformations in linear algebra.