Question

A matrix \(A\) and a vector \(\vec{x}\) are given. Find the product \(A \vec{x}\). $$ A=\left[\begin{array}{cc} 2 & 3 \\ 1 & -1 \end{array}\right], \quad \vec{x}=\left[\begin{array}{l} 4 \\ 9 \end{array}\right] $$

Short answer

The product \( A \vec{x} \) is \( \left[ \begin{array}{c} 35 \\ -5 \end{array} \right] \).

Step-by-step solution

Understand the Matrix and Vector Dimensions

The matrix \( A \) is a 2x2 matrix and the vector \( \vec{x} \) is a 2x1 vector. This means that it is possible to multiply \( A \) by \( \vec{x} \), resulting in a 2x1 vector.

Write the Matrix-Vector Multiplication Formula

The product \( A \vec{x} \) is obtained by taking the dot product of each row of matrix \( A \) with vector \( \vec{x} \). This is calculated as follows: \( A \vec{x} = \left[ \begin{array}{c} 2 \times 4 + 3 \times 9 \ 1 \times 4 + (-1) \times 9 \end{array} \right] \).

Calculate the Dot Product for Each Row

Compute the dot product of the first row of \( A \) with \( \vec{x} \): \[2 \times 4 + 3 \times 9 = 8 + 27 = 35\]Compute the dot product of the second row of \( A \) with \( \vec{x} \): \[1 \times 4 + (-1) \times 9 = 4 - 9 = -5\].

Write the Resulting Vector

Combine the results of the dot products into a vector. Thus, \( A \vec{x} \) becomes:\[ A \vec{x} = \left[ \begin{array}{c} 35 \ -5 \end{array} \right] \].

Key concepts

Matrix-Vector Multiplication

Matrix-vector multiplication is an important operation in linear algebra that combines a matrix and a vector to produce another vector. It is particularly significant in computing linear transformations. When multiplying a matrix by a vector, it is crucial to ensure that the dimensions of the matrix and vector align correctly.

An example of matrix-vector multiplication is with a 2x2 matrix and a 2x1 vector. The matrix, which is composed of rows, is multiplied by the vector, consisting of single-column elements. For each row in the matrix, a dot product is computed with the vector to form each element of the resultant vector.

In the exercise given, the matrix \(A\) is multiplied with a vector \(\vec{x}\), resulting in a new vector \(A\vec{x}\). The elements of \(A\vec{x}\) are determined from the dot products of the rows of matrix \(A\) with \(\vec{x}\). Thus, matrix-vector multiplication effectively applies transformations described by matrix \(A\) to vector \(\vec{x}\).

Dot Product

The dot product is a fundamental operation in vector algebra, used to calculate a single number from two equal-length sequences of numbers. When used in matrix-vector multiplication, it allows each row of a matrix to be multiplied with a vector, producing a single element of a resulting vector.

To calculate the dot product, each element of the row from the matrix is multiplied by the corresponding element of the vector. These products are then summed up to get a single scalar value. In our example, for the first row of the matrix \(A\), the dot product would be computed as \(2 \times 4 + 3 \times 9 = 35\). For the second row, it is \(1 \times 4 + (-1) \times 9 = -5\).

Dot products are particularly useful in various domains such as physics and machine learning, where they help determine projections, compute work done by forces, and many more sensations involving linear computations.

Linear Algebra

Linear algebra is a branch of mathematics that focuses on vectors, vector spaces, and the actions performed upon them. It provides the theoretical foundation for understanding spaces of all dimensions and the linear transformations that act on them.

Key concepts in linear algebra include vectors, matrices, and operations like addition, scalar multiplication, and dot products. The exercise we are analyzing involves these concepts, particularly focusing on matrix-vector multiplication, which is an instance of a linear transformation.

Understanding linear algebra is crucial for fields ranging from computer science to engineering and physics:

• It aids in solving systems of linear equations, crucial for many applications.
• Helps in understanding and manipulating linear transformations.
• Forms the backbone of machine learning algorithms and computer graphics.

Grasping these basic operations and concepts allows for greater insight into the manipulation and understanding of multi-dimensional data.

2x2 Matrix

A 2x2 matrix is a simple and fundamental type of matrix with two rows and two columns. It's an ideal starting point for learning about matrices and operations involving them.

When dealing with a 2x2 matrix, it can represent many useful transformations in two-dimensional space, such as rotations, reflections, and scaling operations. For any 2x2 matrix \(A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\), the key element is its simplicity in calculation yet powerful applications.

In our example, the matrix \(A = \begin{bmatrix} 2 & 3 \ 1 & -1 \end{bmatrix}\) exhibits basic vector transformations when multiplied with vector \(\vec{x}\). Here's why 2x2 matrices are significant:

• They provide insight into understanding more complex matrices.
• Serve as an essential tool in introductory courses of linear algebra.
• Useful in real-world applications requiring simple transformations on a 2D plane.

Mastering operations with 2x2 matrices builds a solid foundation for exploring higher dimensions and more intricate linear transformations.