Question

A matrix \(A\) is given. Sketch \(\vec{x}, \vec{y}, A \vec{x}\) and \(A \vec{y}\) on the same Cartesian axes, where $$\vec{x}=\left[\begin{array}{l} 1 \\ 1 \end{array}\right] \text { and } \vec{y}=\left[\begin{array}{c} -1 \\ 2 \end{array}\right]$$ $$ A=\left[\begin{array}{ll} 1 & 1 \\ 1 & 1 \end{array}\right] $$

Short answer

Sketch vectors from the origin with directions as calculated.

Step-by-step solution

Compute A \(\vec{x}\)

To find \(A \vec{x}\), multiply matrix \(A\) by vector \(\vec{x}\):\[A \vec{x} = \begin{bmatrix} 1 & 1 \ 1 & 1 \end{bmatrix} \begin{bmatrix} 1 \ 1 \end{bmatrix} = \begin{bmatrix} 1\times1 + 1\times1 \ 1\times1 + 1\times1 \end{bmatrix} = \begin{bmatrix} 2 \ 2 \end{bmatrix}\]

Compute A \(\vec{y}\)

Now, calculate \(A \vec{y}\) by multiplying matrix \(A\) by vector \(\vec{y}\):\[A \vec{y} = \begin{bmatrix} 1 & 1 \ 1 & 1 \end{bmatrix} \begin{bmatrix} -1 \ 2 \end{bmatrix} = \begin{bmatrix} 1\times(-1) + 1\times2 \ 1\times(-1) + 1\times2 \end{bmatrix} = \begin{bmatrix} 1 \ 1 \end{bmatrix}\]

Sketch Vectors on Cartesian Axes

On a Cartesian plane, you will draw the following vectors:- \(\vec{x} = \begin{bmatrix} 1 \ 1 \end{bmatrix}\) starting at the origin and pointing to (1,1).- \(\vec{y} = \begin{bmatrix} -1 \ 2 \end{bmatrix}\) starting at the origin and pointing to (-1,2).- \(A \vec{x} = \begin{bmatrix} 2 \ 2 \end{bmatrix}\) starting at the origin and pointing to (2,2).- \(A \vec{y} = \begin{bmatrix} 1 \ 1 \end{bmatrix}\) starting at the origin and pointing to (1,1).The vectors \(A \vec{y}\) and \(\vec{x}\) overlap.

Key concepts

Vector Multiplication

Matrix transformations involve combining matrices and vectors through multiplication. Understanding how vector multiplication works is crucial to grasping matrix transformations.
When you multiply a matrix by a vector, you are effectively applying a linear transformation.
This means you alter the direction and scale of the vector based on the properties of the matrix.

• The multiplication of a matrix by a vector follows specific rules: each entry in the resulting vector is obtained by performing a dot product between the matrix row and the vector column.
• Matrix-vector multiplication transforms the vector components in a consistent manner.

For example, when you computed \( A \vec{x} \), you performed operations between the rows of matrix \( A \) and vector \( \vec{x} \). This directed the resulting vector in a new direction within the plane.

Cartesian Plane

A Cartesian plane is a flat, two-dimensional surface defined by two axes: the x-axis and the y-axis.
When you work with vectors and matrices, visualizing them on the Cartesian plane can help understand their transformations.
Each vector has a specific position and direction in this plane.

• Vectors like \( \vec{x} \) and \( \vec{y} \) are positioned from the origin to a defined point based on their coordinates.
• For example, \( \vec{x} = \begin{bmatrix} 1 \ 1 \end{bmatrix} \) starts at the origin and points to (1, 1) on the Cartesian plane.
• Likewise, after transformation, vectors like \( A \vec{x} \) illustrate how the original vector components change.

By mapping these vectors on the Cartesian plane, one can easily see how transformations through a matrix affect the orientation and position of the vectors.

Matrix-Vector Product

The matrix-vector product is a key operation in matrix algebra that allows us to perform transformations.
It involves the application of a matrix to a vector to produce a new vector.

• The product \( A \vec{x} \) is calculated by summing up the products of corresponding components from the matrix and vector.
• In the given problem, the matrix \( A = \begin{bmatrix} 1 & 1 \ 1 & 1 \end{bmatrix} \) altered the vector \( \vec{x} = \begin{bmatrix} 1 \ 1 \end{bmatrix} \).
• This produced \( A \vec{x} = \begin{bmatrix} 2 \ 2 \end{bmatrix} \), demonstrating a transformation increasing both components proportionally.

The matrix-vector product is particularly significant in representing linear transformations like scaling, rotating, or skewing geometric figures or data points.

Matrix Algebra

Matrix algebra provides the foundation for various transformations and operations involving matrices and vectors. It is a branch of mathematics dealing with matrix-related computations, such as addition, multiplication, and finding inverses.
These operations allow us to solve systems of equations and transform vectors.

• In the problem example, multiplication of matrix \( A \) with vectors \( \vec{x} \) and \( \vec{y} \) showcases a transformation that maintains a pattern based on the matrix's structure.
• Matrix algebra helps explain the geometric transformation properties that each operation holds, whether it's translation, rotation, or scaling.

Understanding these fundamental concepts in matrix algebra facilitates deeper insights into how vector spaces operate, making it vital for students working with linear transformations.