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Chapter 3: Problem 16

\((1,1),(-1,1),(-1,-1)(1,-1)\) are the four corners of a square. Find where they are sent under the action of the matrix $$ \left[\begin{array}{ll} 1 & \frac{1}{2} \\ 0 & 1 \end{array}\right] $$ Plot the given points (in blue) and the new points (in red) in the \(x-y\) plane to see how the matrix shears the square. Do the same for the matrix $$ \left[\begin{array}{cc} -1 & 3 \\ 2 & -1 \end{array}\right] $$

Short Answer

Expert verified
The first matrix transforms points to (1.5,1), (-0.5,1), (-1.5,-1), (0.5,-1). The second matrix transforms them to (2,1), (-4,3), (2,-3), (-2,-1).

Step by step solution

01

Understand the Transformation Matrix

For any given point \((x,y)\) on the plane, when you apply a transformation matrix \(A\), it maps the point to a new point. The new point is found by multiplying the matrix by the position vector \([x \, y]^T\). This means if you have matrix \(A = \begin{pmatrix} a & b \ c & d \end{pmatrix}\), the point \((x, y)\) will transform to \( (ax+by, cx+dy)\).
02

Apply Matrix \(\begin{pmatrix} 1 & \frac{1}{2} \\ 0 & 1 \end{pmatrix}\) Transform

Take each vertex of the square and multiply it by the matrix \(\begin{pmatrix} 1 & \frac{1}{2} \ 0 & 1 \end{pmatrix}\).For \((1,1)\):\(\begin{pmatrix} 1 & \frac{1}{2} \ 0 & 1 \end{pmatrix} \begin{pmatrix} 1 \ 1 \end{pmatrix} = \begin{pmatrix} 1 \times 1 + \frac{1}{2} \times 1 \ 0 \times 1 + 1 \times 1 \end{pmatrix} = \begin{pmatrix} 1.5 \ 1 \end{pmatrix}\)Repeat this for \((-1,1)\), \((-1,-1)\), and \((1,-1)\).
03

Calculate New Points for Matrix \(\begin{pmatrix} 1 & \frac{1}{2} \\ 0 & 1 \end{pmatrix}\)

- For \((-1,1)\):\(\begin{pmatrix} -0.5 \ 1 \end{pmatrix}\)- For \((-1,-1)\):\(\begin{pmatrix} -1.5 \ -1 \end{pmatrix}\)- For \((1,-1)\):\(\begin{pmatrix} 0.5 \ -1 \end{pmatrix}\)
04

Apply Matrix \(\begin{pmatrix} -1 & 3 \\ 2 & -1 \end{pmatrix}\) Transform

Repeat the multiplication process for the second matrix \(\begin{pmatrix} -1 & 3 \ 2 & -1 \end{pmatrix}\) with each original vertex:For \((1,1)\): \(\begin{pmatrix} -1 & 3 \ 2 & -1 \end{pmatrix} \begin{pmatrix} 1 \ 1 \end{pmatrix} = \begin{pmatrix} -1 \times 1 + 3 \times 1 \ 2 \times 1 - 1 \times 1 \end{pmatrix} = \begin{pmatrix} 2 \ 1 \end{pmatrix}\)Follow similarly for \((-1,1)\), \((-1,-1)\), and \((1,-1)\).
05

Calculate New Points for Matrix \(\begin{pmatrix} -1 & 3 \\ 2 & -1 \end{pmatrix}\)

- For \((-1,1)\):\(\begin{pmatrix} -4 \ 3 \end{pmatrix}\)- For \((-1,-1)\):\(\begin{pmatrix} 2 \ -3 \end{pmatrix}\)- For \((1,-1)\):\(\begin{pmatrix} -2 \ -1 \end{pmatrix}\)
06

Visualize the Transformations

Plot the original points \((1,1), (-1,1), (-1,-1), (1,-1)\) in blue. Then, plot the transformed points from each matrix in red. This will illustrate the effects of each matrix on the square's shape.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Matrix Multiplication
Matrix multiplication is a fundamental operation in linear algebra, especially useful when working with transformations. When multiplying a matrix by a vector, each element of the resulting vector is the sum of the products of the corresponding elements of a row of the matrix and the vector. This process shifts and stretches geometric shapes in space.
The basic steps are:
  • Align the matrix and vector in a compatible way such that the number of columns in the matrix matches the number of rows in the vector.
  • Multiply each element of the matrix's row with the corresponding element of the vector.
  • Sum up these products to get an element of the resulting vector.
In our exercise, the transformation matrix is applied to each vertex of the square, resulting in new vertex positions and thus a transformed shape on the coordinate plane.
Coordinate Transformation
Coordinate transformation involves changing the position of a point or a shape on a coordinate plane using a matrix. This matrix acts upon a vector representation of our points, effectively relocating them.
The transformation matrix acts like a machine, conducting operations that systematically reposition points:
  • For each corner of a figure like our square, treat the coordinates as a vector, \(\begin{pmatrix} x \ y \end{pmatrix}\).
  • Multiply this vector by a transformation matrix.
  • The result gives you new coordinates, reflecting a change in location and shape as intended by the matrix's parameters.
Understanding coordinate transformation helps visualize how objects move and change on a plane after applying linear transformations.
Shearing
Shearing is a fascinating transformation that slants the shape of an object. It affects only one dimension of the shape while keeping the length of the sides constant. A typical shearing matrix alters one of the axes while leaving the other unchanged, maintaining parallelism but rotating points.
In shearing transformation:
  • The matrix \(\begin{pmatrix} 1 & \frac{1}{2} \ 0 & 1 \end{pmatrix}\) slants the square to the right.
  • It only modifies the x-coordinate by adding half of the y-coordinate to it, moving points horizontally.
  • The result is a transformed square where each vertex shifts horizontally, but the vertical structure remains aligned.
This simple transformation provides a powerful way to manipulate shapes for various visual effects without altering their fundamental structure.
Geometric Interpretation
The geometric interpretation of matrix transformations provides insight into how mathematical operations transform shapes in space. Visualizing these transformations involves seeing the physical effects on figures like squares or triangles plotted on a coordinate grid.
To understand the geometric changes:
  • Plot the original vertices of a shape, like our square, on an x-y coordinate plane.
  • Apply the designated transformation matrix to each point, recalculating its location.
  • Plot these transformed points to observe the new shape created by the transformation.
In the exercise, the first matrix causes a shearing effect, slightly tilting the square. The second matrix results in a combination of rotation and reflection, substantially altering the square鈥檚 orientation and size. This visualization helps grasp the impact the matrices have, translating numerical operations into tangible changes.

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Most popular questions from this chapter

Let \(A\) and \(B\) be matrices, \(B\) having \(n\) columns, such that \(A B\) exists. Let the matrix \(B\) be thought of as a matrix \(\left[B_{1} B_{2} \ldots B_{n}\right]\) comprising \(n\) column vectors \(B_{i}\), (a) Show that the \(j\) th column of \(A B\) is the matrix \(A B_{j}\). Identify, similarly, the ith row of \(A B\) (b) If \(A\) has a row full of zeros prove that \(A B\) also has such a row. What, if anything, may be said about \(B A\) ? (c) If \(\mathbf{x}\) is the vector $$ \left[\begin{array}{c} x_{1} \\ \vdots \\ x_{n} \end{array}\right] $$ show that \(B \mathrm{x}=B_{1} x_{1}+B_{2} x_{2}+\ldots+B_{n} x_{n}\)

Let \(A\) be the matrix $$ \left[\begin{array}{ll} a & b \\ c & d \end{array}\right] $$ Noting that \((a-x)(d-x)-b c=x^{2}-(a+d) x+a d-b c\), calculate the matrix \(A^{2}-(a+d) A+(a d-b c) I_{2}\). (This is an instance of the celebrated Cayley-Hamilton Theorem.)

The trace \(\operatorname{Tr}(A)\) of the (square) \(n \times n\) matrix \(A\) is defined to be the sum of its diagnonal elements. Prove that, for \(n \times n\) matrices \(A\) and \(B\), we have $$ \operatorname{Tr}(A+B)=\operatorname{Tr}(A)+\operatorname{Tr}(B) \text { and } \operatorname{Tr}(A B)=\operatorname{Tr}(B A) $$ Use these results to deduce that there are no \(n \times n\) matrices \(A\) and \(B\) such that \(A B-B A=I_{n}\)

For vectors \(\mathbf{v}_{1}=\left(a_{1}, b_{1}, c_{1}\right), \mathbf{v}_{2}=\left(a_{2}, b_{2}, c_{2}\right)\) in \(\mathbb{R}^{3}\) we define their vector product to be the vector \(\mathbf{v}_{1} \wedge \mathbf{v}_{2}=\left(b_{1} c_{2}-b_{2} c_{1}, c_{1} a_{2}-c_{2} a_{1}, a_{1} b_{2}-a_{2} b_{1}\right) .\) Show that (i) \(\mathbf{v}_{1} \wedge \mathbf{v}_{2}=-\left(\mathbf{v}_{2} \wedge \mathbf{v}_{1}\right)\), (ii) \(\mathbf{v}_{1} \wedge\left(\mathbf{v}_{1} \wedge \mathbf{v}_{2}\right)=\mathbf{v}_{2} \wedge\left(\mathbf{v}_{1} \wedge \mathbf{v}_{2}\right)=(0,0,0)\left\\{\right.\) so that \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2}\) are each orthogonal to \(\left.\mathbf{v}_{1} \wedge \mathbf{v}_{2}\right\\}\) and find particular \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2}\) such that (iii) \(\left(\mathbf{v}_{1} \wedge \mathbf{v}_{2}\right) \wedge \mathbf{v}_{2} \neq \mathbf{v}_{1} \wedge\left(\mathbf{v}_{2} \wedge \mathbf{v}_{2}\right)\)

(a) Find \(2 \times 2\) matrices \(A\) and \(B\) such that \((A B)^{2} \neq A^{2} B^{2}\). (b) Find \(2 \times 2\) matrices \(C\) and \(D\) such that \(C D \neq D C\) and yet \((C D)^{2}\) is equal to \(C^{2} D^{2}\).
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