Question

A matrix \(A\) and a vector \(\vec{x}\) are given. Find the product \(A \vec{x}\). $$ A=\left[\begin{array}{lll} 1 & 2 & 3 \\ 1 & 0 & 2 \\ 2 & 3 & 1 \end{array}\right], \quad \vec{x}=\left[\begin{array}{l} x_{1} \\ x_{2} \\ x_{3} \end{array}\right] $$

Short answer

\( A\vec{x} = \begin{bmatrix} 1x_1 + 2x_2 + 3x_3 \\ 1x_1 + 0x_2 + 2x_3 \\ 2x_1 + 3x_2 + 1x_3 \end{bmatrix} \)

Step-by-step solution

Set Up the Problem

We are given a matrix \( A \) and a vector \( \vec{x} \). The matrix \( A \) is a 3x3 matrix:\[A = \begin{bmatrix} 1 & 2 & 3 \ 1 & 0 & 2 \ 2 & 3 & 1 \end{bmatrix}\]The vector \( \vec{x} \) is a 3x1 column vector:\[\vec{x} = \begin{bmatrix} x_1 \ x_2 \ x_3 \end{bmatrix}\]We are asked to find the product \( A\vec{x} \).

Calculate Each Element of the Resulting Vector

The product \( A \vec{x} \) is computed by multiplying each row of the matrix \( A \) with the vector \( \vec{x} \), resulting in a new vector:- First element: \( 1 \cdot x_1 + 2 \cdot x_2 + 3 \cdot x_3 \)- Second element: \( 1 \cdot x_1 + 0 \cdot x_2 + 2 \cdot x_3 \)- Third element: \( 2 \cdot x_1 + 3 \cdot x_2 + 1 \cdot x_3 \)

Write the Resulting Vector

Combine the expressions for the elements to write the resulting vector:\[A\vec{x} = \begin{bmatrix} 1x_1 + 2x_2 + 3x_3 \ 1x_1 + 0x_2 + 2x_3 \ 2x_1 + 3x_2 + 1x_3 \end{bmatrix}\]

Simplify the Expression (if needed)

Note that each element is already in its simplest form. The resulting vector can be expressed using the individual components or by evaluating specific values of \( x_1, x_2, \text{ and } x_3 \). For the given answer, we're keeping them in terms of their variables.

Key concepts

Linear Algebra

Linear algebra is a branch of mathematics that deals with vectors, matrices, and linear transformations. It’s fundamental in understanding and solving equations that involve linear relationships. In the current exercise, linear algebra principles help us to systematically handle multiple equations using matrix operations.

One of the key ideas in linear algebra is the concept of a linear equation, which can be thought of as a balance or equality involving a linear combination of variables. A set of such equations can be expressed concisely using matrices and vectors, as shown in our exercise. Here, the relationship is expressed in a matrix-vector form:

• Matrix: A grid-like array of numbers, symbols, or expressions arranged in rows and columns.
• Vector: A mathematical object with magnitude and direction, often representing a point or a line in space.

These concepts are not just abstract ideas; they underpin many real-world applications such as computer graphics, engineering simulations, and data science.

Matrix Operations

Matrix operations are fundamental techniques used in linear algebra that allow us to manipulate matrices to achieve desired results. The multiplication of a matrix by a vector is one of the core operations, also known as matrix-vector multiplication.

The process involves taking a matrix and a vector and yielding a new vector by carrying out a series of dot products. Each element of the resulting vector is calculated by:

• Taking one row of the matrix
• Multiplying each element of that row by the corresponding element of the vector
• Summing these products to obtain a single number

The important points to remember here are:

• The number of columns in the matrix must match the number of entries in the vector.
• Each resulting vector element corresponds to a row in the matrix.

This method of matrix-vector multiplication is essential in transforming data and solving systems of equations, providing a streamlined and efficient process.

Vector Space

In linear algebra, a vector space is a collection of vectors that can be added together and multiplied by scalars while still remaining within the same space. It provides a framework to discuss vectors irrespective of their magnitude or direction.

The concept of vector spaces is crucial in understanding vector operations and matrix transformations. In the exercise provided, the interactions between matrix and vector showcase how transformation works:

• The vector \( \vec{x} \) is part of a vector space, and any linear combination of its elements remains in that space.
• The matrix \( A \) acts as a transformation tool, changing or moving vectors within the space.

This means that any operation with the matrix and vector combination will yield another vector within the same space.
Understanding vector spaces is key to recognizing how data sets, physics problems, and spatial transformations can all be viewed and resolved through linear algebra. By manipulating vector spaces, we solve complex problems efficiently using basic rules of vector addition and scalar multiplication.