#vector geometry

Self teaching from this very overpriced textbook yay

Finished Chapter 3: Vectors in ℝ2 and ℝ3 

Questions, comments, corrections? Message here or on the blog!

Finished Chapter 5: Linear Transformations!

Finished Chapter 4: Vector Spaces.

Special examples of linear transformations in ℝ² include rotations and reflections.

Geometric descriptions of vector addition and scalar multiplication are used to show that a rotation of vectors through an angle and that a reflection of a vector across a line are examples of linear transformations.

Theorem of Rotation Consider the following diagram:

To obtain T(u + v), T(u) and T(v) are added. This is because, if you add T(u) to T(v), the result is the diagonal of the parallelogram determined by T(u) and T(v). If vectors u and v were added first and then apply the transformation T to the u + v vector, the result is T(u + v), which is the same vector as T(u) + T(v). Therefore, T(u + v) = T(u) + T(v).

This is because the rotation by T preserves all angles between the vectors. It preserves the shape of the parallelogram determined by the two vectors. T distributes across addition of the vectors in ℝ².

If k is a scalar, then T(ku) = kT(u). Then not only does the rotation by T preserve all angles between the vector, it also preserves their magnitude.

Therefore, rotations are an example of a linear transformation, where a matrix of a linear transformation can be found that rotates all vectors through an angle of θ.

Let Rθ : ℝ² -> ℝ² be a linear transformation given by rotating vectors through an angle of θ. Then the matrix A of Rθ is the following:

The following is the function of Rθ for x ∈ ℝ²:

Rθ(x) = Ax

where A is the matrix defined above.

Let e₁ = [1 0]T and e₂ = [0 1]T. These are the geometric vectors which lie on the positive x-axis and the positive y-axis.

Let Rπ/2 : ℝ² -> ℝ² denote the rotation through π/2.

To find the matrix of Rπ/2, use the theorem of the matrix that induces Rπ/2.

To find Rπ/2(x) where x = [1 -2]T, use the function of Rθ.

Theorem of Reflection Let Qm : ℝ² -> ℝ² be a linear transformation given by reflecting vectors over the line y = mx. Then matrix of Qm is the following:

The following is the function of Qm for x ∈ ℝ²:

Qm(x) = Ax

where A is the matrix defined above.

Let Q₂ : ℝ² -> ℝ² denote the reflection over the line y = 2x. Then Q₂ is a linear transformation.

To find the matrix of Q₂, use the theorem of the matrix that induces Q₂.

To find Q₂(x) where x = [1 -2]T, use the function of Qm.

Rotation and Reflection When finding the matrix of the composite transformation Qm ○ Rθ, where matrix A is the matrix of Rθ and matrix B is the matrix of Qm, the matrix of the composite transformation Qm ○ Rθ is BA.

To find the matrix of the composite transformation obtained by first rotating all vectors through an angle of π/6 and then reflecting through the x-axis, start by finding the matrix A of Rπ/6.

Next, find the matrix B of Qm. Since the vectors are reflected through the x-axis, m = 0.

By definition, the matrix of the composite transformation Q₀ ○ Rπ/6 is BA.

The action of the linear transformation was to multiply by a matrix A. This is always the case for linear transformations.

If T is any linear transformation that maps ℝⁿ to ℝᵐ, there is always an m x n matrix A with the property that makes the following true:

T(x) = Ax for all x ∈ ℝⁿ

Informal Theorem of the Matrix of a Linear Transformation Let T : ℝⁿ -> ℝᵐ be a linear transformation. Then the matrix A can be found, where T(x) = Ax. Then T is determined, or induced, by the matrix A.

Suppose T : ℝⁿ -> ℝᵐ is a linear transformation, and the matrix A that is defined by this linear transformation is desired to be found.

Note that the following is true:

eᵢ is the n x 1 vector which equals the ith column of the identity matrix In, where this vector has all zero entries except for the ith entry.

Since T is linear, the following is true:

Therefore, the desired matrix is obtained from constructing the ith column as T(eᵢ).

Formal Theorem of the Matrix of a Linear Transformation Let T : ℝⁿ -> ℝᵐ be a linear transformation. Then the matrix A that satisfies T(x) = Ax is defined as the following:

where eᵢ is the ith column of In and so T(eᵢ) is the ith column of matrix A.

Corollary of Matrix and Linear Transformation A transformation T is a linear transformation if and only if it is a matrix transformation.

Suppose T : ℝ³ -> ℝ² is a linear transformation, where the following is true:

To find the matrix A of T such that T(x) = Ax for all x, construct the matrix A.

Matrix A will be a 2 x 3 matrix so that a 3 x 1 vector transforms into a 2 x 1 vector.

To complete matrix A, the columns T(e₁), T(e₂), and T(e₃) must be found. The question gives these three columns already, because they are all columns from the identity matrix I₃.

This example had already given the resulting vectors of T(e₁), T(e₂), and T(e₃), so constructing matrix A was simple, because these vectors can be used as the columns of matrix A.

The follow example will show how to find matrix A when T(eᵢ) isn't given clearly.

Suppose T : ℝ² -> ℝ² is a linear transformation, where the following is true:

To find the matrix A of the transformation T such that T(x) = Ax for all x, the action of T on e₁ and e₂ needs to be determined.

For e₁, suppose there exists x and y such that the following is true:

Since T is linear, the following is true:

Substituting the T functions:

The x and y values needs to be found in order to find the first column of the matrix A, T(e₁).

To find x and y, solve the system, which gives the following solution:

x = 1 x - y = 0

Therefore, x = 1 and y = 1 is the solution to the system.

Substituting these values into the substituted T function equation:

Therefore, the first column of the matrix A is [4 4]T.

Computing the second column of matrix A uses the same procedure. The resulting matrix A is the following:

This example demonstrates a long procedure for finding the matrix of A. This method is reliable and will always result in the correct matrix A, however the the following procedure provides an alternative method.

Procedure of Finding the Matrix A of an Inconveniently Defined Linear Transformation Recall that [A₁ ... An] denotes a matrix where for each ith column has the vector Aᵢ.

Suppose T : ℝⁿ -> ℝᵐ is a linear transformation. Also suppose there exists a set of vectors {a₁, ..., an} in ℝⁿ such that the inverse of the column matrix [a₁ ... an]⁻¹ exists, and the following is true:

T(aᵢ) = bᵢ

Then the matrix of T must be of the following form:

[b₁ ... bn][a₁ ... an]⁻¹

Suppose T : ℝ³ -> ℝ³ is a linear transformation, where the following is true:

To find the matrix C of this linear transformation, the function T(aᵢ) = bᵢ is used, because eᵢ is not given clearly as columns of the 3 x 3 identity matrix.

Using aᵢ to create the matrix A:

Using bᵢ to create the matrix B:

According to the procedure for finding the matrix C of a linear transformation,  C = BA⁻¹, where A⁻¹ is the inverse matrix of A.

After finding the inverse of matrix A and multiplying this with matrix B, matrix C of T is obtained.

Matrix of a Projection Map Consider the map of vector v -> proju(v). This map is linear, because of the properties of the dot product.

Therefore, T(v) = proju(v) is also linear and the ith column of the matrix of T is defined as the following:

T(eᵢ) = proju(eᵢ)

Let u = [1 2 3]T and let T be the projection map T : ℝ³ -> ℝ³, defined by T(v) = proju(v) for any v ∈ ℝ³.

To determine if this transformation comes from multiplication by a matrix, it is seen that T(v) = proju(v) is linear, so matrix A such that T(x) = Ax can be found.

To find the matrix A of T, the columns of matrix A are defined as T(eᵢ). Since T(eᵢ) = proju(eᵢ) gives the ith column of matrix A, the following needs to be computed:

Since matrix A will be 3 x 3, e₁ = [1 0 0]T, e₂ = [0 1 0]T, and e₃ = [0 0 1]T. Computing proju(eᵢ) for each e₁, e₂, and e₃ yields the following columns of matrix A:

Therefore, the desired matrix A of T is the following:

Definition of Orthogonal Matrices For some special matrices, its transpose equals its inverse. When an n x n matrix has all real entries and its transpose equals its inverse, the matrix is called an orthogonal matrix.

A real n x n matrix U is called an orthogonal matrix if the following is true:

UUT = UTU = I

Given the following matrix U:

To show it is an orthogonal matrix, verify it using one of the equations from the definition.

Since UUT = I, this matrix is orthogonal.

Given the following matrix U:

To show it is an orthogonal matrix, verify it using one of the equations from the definition.

Since UTU = I, this matrix is orthogonal.

Orthogonal Matrices and Summation Notation When the matrix U is orthogonal, the following is true:

The product of the ith row of U with the kth row of its transpose equals to 1 if i = k, or it equals 0 if i ≠ k. The same is true for the columns, because UTU = I.

The product of the ith column of U with the kth column of its transpose equals to 1 if i = k, or it equals 0 if i ≠ k.

If u₁, ..., un are the columns of an orthogonal matrix U, then the following is true:

The columns and rows form an orthonormal set of vectors. Therefore, a matrix is orthogonal if its rows or columns form an orthonormal set of vectors.

Note that the convention is to call such a matrix orthogonal instead of orthonormal.

Proposition of Orthonormal Basis The rows of an n x n orthogonal matrix form an orthonormal basis of ℝⁿ.

Any orthonormal basis of ℝⁿ can be used to construct an n x n orthogonal matrix.

Proposition of Determinant of Orthogonal Matrices Suppose U is an orthogonal matrix. Then its determinant is the following:

det(U) = ±1

The following is the proof:

Proper and Improper Orthogonal Matrices Orthogonal matrices are divided into two classes, proper and improper.

The proper orthogonal matrices are the orthogonal matrices with its determinant equal to 1.

The improper orthogonal matrices are the orthogonal matrices with its determinant equal to -1.

The reason for the classes is that the improper orthogonal matrices are sometimes considered to have no physical significance. These matrices cause a change in orientation, corresponding to material passing through itself in a non physical manner.