Question

A matrix \(A\) and a vector \(\vec{x}\) are given. Find the product \(A \vec{x}\). $$ A=\left[\begin{array}{cc} -1 & 4 \\ 7 & 3 \end{array}\right], \quad \vec{x}=\left[\begin{array}{c} 2 \\ -1 \end{array}\right] $$

Short answer

The product \( A \vec{x} \) is \( \left[\begin{array}{c}-6 \\ 11\end{array}\right] \).

Step-by-step solution

Understand the Matrix-Vector Product

The product of a matrix \( A \) and a vector \( \vec{x} \) is a new vector where each element is the dot product of the corresponding row of the matrix with the vector. For a matrix \( A \) of size \( m \times n \) and a vector \( \vec{x} \) with \( n \) elements, the result will be a vector of size \( m \). Our matrix \( A \) is \( 2 \times 2 \), and our vector \( \vec{x} \) has 2 elements, so the result will be a vector with 2 elements.

Calculate the First Element of the Result Vector

The first element of the product vector is obtained by calculating the dot product of the first row of \( A \) with \( \vec{x} \). The first row of \( A \) is \([-1, 4]\), and \( \vec{x} \) is \([2, -1]\).\[ ext{First element} = (-1) \times 2 + 4 \times (-1) = -2 - 4 = -6\]

Calculate the Second Element of the Result Vector

Similarly, calculate the second element of the product vector using the second row of \( A \). The second row of \( A \) is \([7, 3]\).\[ ext{Second element} = 7 \times 2 + 3 \times (-1) = 14 - 3 = 11\]

Construct the Resulting Product Vector

Combine the results from Step 2 and Step 3 to form the resulting vector:\[A \vec{x} = \left[\begin{array}{c}-6 \11\end{array}\right]\]

Key concepts

Dot Product

The dot product is a fundamental concept in linear algebra that helps us evaluate the result of multiplying a row vector by a column vector. Imagine two vectors, one with values \(a_1, a_2, \ldots, a_n\) and another with \(b_1, b_2, \ldots, b_n\). To find their dot product, we multiply each pair of corresponding components and sum up these products: \[ (a_1 \cdot b_1) + (a_2 \cdot b_2) + \ldots + (a_n \cdot b_n) \] Think of it as a way to combine two different sets of information into a single value, summarizing how much these two sets "agree" with each other. The dot product is useful when calculating the result of matrix-vector multiplication because each element of the resulting vector involves the dot product of a row from the matrix and the given vector.

Matrix Operations

Matrix operations are procedures that involve multiple entities in linear algebra, such as rows, columns, and numbers within a matrix. Matrix-vector multiplication is an essential operation where each row of a matrix is engaged in a dot product with a vector to create a new vector. Here’s how each step proceeds:

• Take one row of the matrix
• Perform a dot product with the entire vector
• Write down the result as a component of the new vector

This continues until all rows are processed, resulting in a new vector. In our case, with matrix \(A\) being 2x2 and vector \(\vec{x}\) having two elements, the result is a 2-component vector. Matrix operations provide a structured approach to handling data, making it easier to solve systems of linear equations, transition between coordinate spaces, and apply transformations.

Linear Algebra Concepts

Linear algebra is a branch of mathematics focusing on vectors, matrices, and their interactions. These interactions include operations like addition, multiplication, and transformation. At its core, linear algebra revolves around solving problems related to linear equations and mappings, which are foundational in many fields such as computer science, engineering, and physics.

• Vectors: Represented typically as arrows that have direction and magnitude.
• Matrices: Collections of numbers organized into rows and columns, used to solve systems of linear equations.
• Transformations: Operations that alter the form of vectors, such as rotation, scaling, or translation.

These concepts might seem abstract initially, but they become practical and intuitive as you grasp their use in computing, simulations, and various scientific modeling tasks. Understanding how to handle matrix-vector multiplication is a stepping stone to mastering more complex linear algebra applications.