Question

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

Short answer

The product \(A \vec{x}\) results in the vector \([2x_1 - x_2, 4x_1 + 3x_2]^T\).

Step-by-step solution

Define Matrix-Vector Multiplication

The product of a matrix \(A\) and a vector \(\vec{x}\) is a new vector, where each component is a linear combination of the components of \(\vec{x}\) using the rows of \(A\).

Compute First Component of Product

Take the first row of matrix \(A\), which is \([2, -1]\), and multiply it by the vector \(\vec{x} = [x_1, x_2]\). Compute the dot product: \(2x_1 - 1x_2 = 2x_1 - x_2\).

Compute Second Component of Product

Take the second row of matrix \(A\), which is \([4, 3]\), and multiply it by the vector \(\vec{x} = [x_1, x_2]\). Compute the dot product: \(4x_1 + 3x_2\).

Write the Resultant Vector

The result is a vector formed by the components from the previous steps: \[A \vec{x} = \begin{bmatrix} 2x_1 - x_2 \ 4x_1 + 3x_2 \end{bmatrix}\].

Key concepts

Linear Combinations

In matrix-vector multiplication, the idea of a linear combination plays a crucial role. To understand it simply, think of a linear combination as a blend of vectors where each vector is multiplied by a corresponding scalar (number). Here’s how it applies to our task of multiplying matrix \( A \) by vector \( \vec{x} \):

• Each entry in the resulting product vector \( A \vec{x} \) is formed by a linear combination of the components of \( \vec{x} \).
• The coefficients of this combination are given by the elements of the rows of matrix \( A \).

Break the process down as follows:
For the first component, you take \( 2x_1 - x_2 \), which is a linear combination using the first row of \( A \) and vector \( \vec{x} \).
Similarly, for the second component, \( 4x_1 + 3x_2 \) is another linear combination, this time using the second row of \( A \).
Through these linear combinations, matrix-vector multiplication elegantly blends together the elements of the vector using the "instructions" from the matrix.

Dot Product

The dot product, also known as the scalar product, is another fundamental concept involved in matrix-vector multiplication. It helps to succinctly describe how to calculate each element of the resulting vector from multiplying a matrix by a vector.

• A dot product is a specific type of multiplication that operates on two vectors and returns a single number, or scalar.
• Each component of the product vector \( A \vec{x} \) results from the dot product of a row from \( A \) and vector \( \vec{x} \).

Here’s the step-by-step of how this is done:
For the first entry, you take the first row \( [2, -1] \) from matrix \( A \) and compute its dot product with vector \( \vec{x} = [x_1, x_2] \). It precisely results in \( 2x_1 - x_2 \).
The second entry stems from the dot product of the second row \( [4, 3] \) and the vector \( \vec{x} \), resulting in \( 4x_1 + 3x_2 \).
By understanding the dot product process, you can easily compute any component of the matrix-vector product.

Matrix Algebra

Matrix algebra provides a set of rules and operations, including addition, subtraction, and multiplication, specific to matrices and vectors. This framework is vital for solving problems like finding \( A \vec{x} \), which can come up frequently in fields such as engineering, computer science, and physics.

• Matrix-vector multiplication is one operation within matrix algebra that combines a matrix and a vector to produce another vector.
• This operation is a foundation block for more complex matrix algebra tasks such as solving systems of equations or performing transformations.

In our problem, using matrix algebra:
You took matrix \( A \), organized by its rows \([2, -1]\) and \([4, 3]\), and applied it to vector \( \vec{x} \).This resulted in a new vector, precisely calculated using the principles of linear combinations and dot products.
Matrix algebra is a powerful tool that abstracts and simplifies transformations and operations involving multiple variables, effectively serving as a universal language for linear equations and transformations.