Question

A matrix \(A\) is given. Sketch \(\vec{x}, \vec{y}, A \vec{x}\) and \(A \vec{y}\) on the same Cartesian axes, where $$\vec{x}=\left[\begin{array}{l} 1 \\ 1 \end{array}\right] \text { and } \vec{y}=\left[\begin{array}{c} -1 \\ 2 \end{array}\right]$$ $$ A=\left[\begin{array}{cc} 1 & -1 \\ 2 & 3 \end{array}\right] $$

Short answer

A transformation of \(\vec{x}\) results in \(A\vec{x} = \begin{bmatrix} 0 \\ 5 \end{bmatrix}\) and \(\vec{y}\) in \(A\vec{y} = \begin{bmatrix} -3 \\ 4 \end{bmatrix}\). Plot these on the Cartesian plane.

Step-by-step solution

Calculate A\vec{x}

Given \(\vec{x} = \begin{bmatrix} 1 \ 1 \end{bmatrix}\) and \(A = \begin{bmatrix} 1 & -1 \ 2 & 3 \end{bmatrix}\), we need to calculate \(A\vec{x}\). This is done by performing matrix multiplication: \[ A\vec{x} = \begin{bmatrix} 1 & -1 \ 2 & 3 \end{bmatrix} \begin{bmatrix} 1 \ 1 \end{bmatrix} = \begin{bmatrix} (1)(1) + (-1)(1) \ (2)(1) + (3)(1) \end{bmatrix} = \begin{bmatrix} 0 \ 5 \end{bmatrix} \].So, \(A\vec{x} = \begin{bmatrix} 0 \ 5 \end{bmatrix}\).

Calculate A\vec{y}

Next, let's find \(A\vec{y}\), where \(\vec{y} = \begin{bmatrix} -1 \ 2 \end{bmatrix}\). Using matrix multiplication again:\[ A\vec{y} = \begin{bmatrix} 1 & -1 \ 2 & 3 \end{bmatrix} \begin{bmatrix} -1 \ 2 \end{bmatrix} = \begin{bmatrix} (1)(-1) + (-1)(2) \ (2)(-1) + (3)(2) \end{bmatrix} = \begin{bmatrix} -1 - 2 \ -2 + 6 \end{bmatrix} = \begin{bmatrix} -3 \ 4 \end{bmatrix} \].So, \(A\vec{y} = \begin{bmatrix} -3 \ 4 \end{bmatrix}\).

Sketch \vec{x} and \vec{y} on Cartesian Plane

On a Cartesian coordinate system, plot \(\vec{x} = \begin{bmatrix} 1 \ 1 \end{bmatrix}\) which is the point (1,1). Then plot \(\vec{y} = \begin{bmatrix} -1 \ 2 \end{bmatrix}\) which is the point (-1,2). These represent the vectors \(\vec{x}\) and \(\vec{y}\) originating from the origin.

Sketch A\vec{x} and A\vec{y} on Cartesian Plane

Similarly, plot the calculated vectors from Steps 1 and 2. Place \(A\vec{x} = \begin{bmatrix} 0 \ 5 \end{bmatrix}\) at the point (0,5) and \(A\vec{y} = \begin{bmatrix} -3 \ 4 \end{bmatrix}\) at the point (-3,4). These vectors also originate from the origin and show how the matrix transformation \(A\) affects original vectors \(\vec{x}\) and \(\vec{y}\).

Key concepts

Matrix Multiplication

Matrix multiplication is a crucial tool in transforming vectors or generating new matrices. It involves multiplying rows of one matrix by columns of another. In this exercise, we used matrix multiplication to transform vectors \( \vec{x} \) and \( \vec{y} \) using the matrix \( A \). This enabled us to find transformed vectors \( A\vec{x} \) and \( A\vec{y} \).

For example, the matrix product \( A\vec{x} \) was calculated by:

• Take the first row of \( A \) and multiply it by \( \vec{x} \)'s column: \((1 \times 1) + (-1 \times 1) = 0\).
• Then, take the second row of \( A \) and do the same: \((2 \times 1) + (3 \times 1) = 5\).

This gives us the vector \( \begin{bmatrix} 0 \ 5 \end{bmatrix} \). This process shows how each element in the result comes from specific combinations of the original rows and columns.

Matrix multiplication gives a new dimension to vector manipulation, allowing us to rotate, scale, and transform vectors using matrices.

Cartesian Plane

The Cartesian plane is a two-dimensional coordinate system that allows us to visualize mathematical concepts such as vectors. It consists of two perpendicular axes - the horizontal (x-axis) and vertical (y-axis). Every point on this plane can be expressed as an ordered pair \((x, y)\).

In our exercise, we plotted vectors \( \vec{x} \) and \( \vec{y} \) on the Cartesian plane to see their positions. Vector \( \vec{x} \) was depicted as the point \((1, 1)\), while \( \vec{y} \) was plotted at \((-1, 2)\).

Once multiplied by matrix \( A \), the new vectors \( A\vec{x} \) and \( A\vec{y} \) were plotted at \((0, 5)\) and \((-3, 4)\) respectively. By visualizing these vectors on the Cartesian plane, it becomes easier to understand how transformations affect their positions.

The Cartesian plane thus serves as a valuable tool for plotting and analyzing vector transformations, offering a clear visual representation of the mathematical changes.

Vector Representation

Vectors are mathematical objects that have both magnitude and direction. In the context of the Cartesian plane, each vector can be represented by a point \((x, y)\). Vectors like \( \vec{x} = \begin{bmatrix} 1 \ 1 \end{bmatrix} \) translate to points on the plane corresponding to their components.

The transformation of these vectors via a matrix \( A \) changes their direction and magnitude, resulting in new vector representations \( A\vec{x} \) and \( A\vec{y} \). For instance, \( \vec{x} \) shifts from \((1, 1)\) to \( A\vec{x} = (0, 5)\).

Understanding vector representation is key to grasping how scalar multiplication, addition, and various transformations can reshape a vector. By seeing how the original vectors \( \vec{x} \) and \( \vec{y} \) evolve into \( A\vec{x} \) and \( A\vec{y} \), we can appreciate the power of vectors in modeling physical and spatial phenomena.

Vectors provide a meaningful way to represent quantities that inherently have directionality, making them vital in fields like physics and computer graphics.