Question

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

Short answer

\( A\vec{x} \) and \( A\vec{y} \) both equal \( \begin{bmatrix} 3 \\ -3 \end{bmatrix} \) and plot to the same point.

Step-by-step solution

Calculate \( A\vec{x} \)

First, we need to compute the result of applying matrix \( A \) to vector \( \vec{x} \). Using matrix multiplication rules:\[ A\vec{x} = \begin{bmatrix} 1 & 2 \ -1 & -2 \end{bmatrix} \begin{bmatrix} 1 \ 1 \end{bmatrix} = \begin{bmatrix} 1(1) + 2(1) \ -1(1) - 2(1) \end{bmatrix} = \begin{bmatrix} 3 \ -3 \end{bmatrix} \]

Calculate \( A\vec{y} \)

Now, we calculate the result of applying matrix \( A \) to vector \( \vec{y} \):\[ A\vec{y} = \begin{bmatrix} 1 & 2 \ -1 & -2 \end{bmatrix} \begin{bmatrix} -1 \ 2 \end{bmatrix} = \begin{bmatrix} 1(-1) + 2(2) \ -1(-1) - 2(2) \end{bmatrix} = \begin{bmatrix} 3 \ -3 \end{bmatrix} \]

Plot the vectors

On a Cartesian coordinate system, plot the following vectors:- \( \vec{x} = \begin{bmatrix} 1 \ 1 \end{bmatrix} \) - \( \vec{y} = \begin{bmatrix} -1 \ 2 \end{bmatrix} \) - \( A\vec{x} = A\vec{y} = \begin{bmatrix} 3 \ -3 \end{bmatrix} \)First plot \( \vec{x} \) starting from the origin to the point (1, 1). Then plot \( \vec{y} \) from the origin to (-1, 2). Finally, draw both \( A\vec{x} \) and \( A\vec{y} \) from the origin to (3, -3), noticing they coincide at the same point.

Key concepts

Vector Transformation

Vector transformation is a core concept in linear algebra that involves using matrices to manipulate vectors. In the exercise, we see this through the application of the matrix \( A \) to the vectors \( \vec{x} \) and \( \vec{y} \). Consider each vector as a point in space. When a matrix like \( A \) acts on a vector, it transforms it in a certain way:

• It can change the direction.
• It can scale the vector, making it longer or shorter.
• It can rotate the vector around the origin.

In this specific case, applying \( A \) to both \( \vec{x} \) and \( \vec{y} \) results in the same outcome, \( A\vec{x} = A\vec{y} = \begin{bmatrix} 3 \ -3 \end{bmatrix} \). This outcome implies a transformation where different original vectors can end up at the same transformed vector. Here, transforming means effectively applying a rule represented by the matrix to systematically modify the vectors.

Cartesian Coordinates

Cartesian coordinates provide a way to represent vectors in a plane using two axes: the horizontal (x-axis) and the vertical (y-axis). Each vector in the plane can be described by its location in terms of these axes. In the given exercise:

• Vector \( \vec{x} = \begin{bmatrix} 1 \ 1 \end{bmatrix} \) signifies a point that lies in the first quadrant of the coordinate system.
• Vector \( \vec{y} = \begin{bmatrix} -1 \ 2 \end{bmatrix} \) is located in the second quadrant.

When we graph these vectors, we start from the origin, which is the point \( (0,0) \) on the graph. We compare where the tip of each vector lands on the grid:

• \( \vec{x} \) approaches point (1, 1).
• \( \vec{y} \) reaches (-1, 2).
• After transformation, both result in (3, -3).

Understanding how vectors occupy spaces within these axes is critical for visually interpreting mathematics and engineering tasks.

Linear Algebra

Linear algebra is a field of mathematics focusing on vector spaces and linear mappings between them. It forms the backbone for matrix operations, such as those seen in our exercise. At its core, linear algebra deals with concepts such as:

• Vectors and matrices.
• Transformations and their properties.
• Solving systems of linear equations.

Matrix multiplication, like in the calculation of \( A\vec{x} \) or \( A\vec{y} \), is central to linear algebra. It involves applying the matrix to a vector, resulting in a new vector. This operation is governed by specific rules like:

• Each element of the resulting vector is a dot product of a row of the matrix and the input vector.
• The size compatibility requirement, meaning the number of columns in the matrix must match the number of rows in the vector.

In our example:

• The initial transformation illustrates how these rules create new, equivalent vectors from different starting points due to shared transformation properties.

Mastering linear algebra allows one not only to solve such problems but to understand how systems work when masses of data are dealt with simultaneously.