Description

T R2×2
Let the vectors obtained after rotation be a′,b′. Then we have:
a′
b′ = T(θ)
⟹ d
A =
To find the first column, we give an input vector with all zeros except the first element, which we set as 1. Then, the output vector will be the first column of the matrix A.
x = [1 0 0 … 0]
⟹ Ax = [a11 a21 a31 … am1 ]T = y
We repeat this process with all the other columns to find the matrix A. To find the second column of the matrix A, we use the input vector as:
x = … 0]T
Using Ax = y we get the second column of the matrix
We repeat this process until we get all the columns of the matrix A
a) Let x ∈ Rn, We can represent x as
x = y
= y ynbn
= [ y
= B
where B is the matrix with columns as the basis vectors , n and y is the coordinate vector of x in basis B.
Similarly, we can write that
x
]z
where V is the matrix with columns as the basis vectors , n and z is the coordinate vector of x in basis V.
Now, we have defined a transformation T : Rn → = z.
To prove that T is a linear transformation, we consider vectors and y2, which are the coordinate vectors of x1 and x2 in the basis B respectively. Similarly, consider z1 which are the coordinate vectors of x1 and x2 in the basis V respectively.
⟹
⟹
⟹ B(αy1 + z2) ∀α,β ∈ R
Since T(y) = z, we have:
⟹ T(αy1 + βy
T(y2) ∀α,β ∈ R
Thus, T is a linear transformation.
b) As T is a linear transformation, we can associate a matrix with it such that T(y) = Ay = z.
v1,v2,…,vn are basis vectors which form the columns of the matrix B. Basis vectors are by definition, independent.
Hence, the columns of the matrix obtained above are also independent. Therefore, B is invertible.
⟹ z = V−1By
⟹ T(y) = V−1By
B
Thus, the matrix A is the change of basis matrix from
Let x ∈ V.
x = c1v1 + c2v2 + ⋯cnvn
Thus, the coordinate vector of x is
⎤
⎦
T(x) = T(c1v1 + c2v2 + ⋯cnvn)
As matrices are linear transformations by definition:
T
Now, T(v1) can be represented in basis W as
T(v1) = a11w1 + a21w2 + ⋯am1wm
Similarly,
T(v2) = a12w1 + a22w2 + ⋯am2wm
And so on, until:
T(vn) = a1nw1 + a2nw2 + ⋯amnwm
Therefore,
T amnwm)
⎡ a ⋯ a1n ⎤⎡c1⎤
a ⋯ a2n c2
⟹ T(y)basis W Ay
⎣a ⋯ amn ⎦⎣cn⎦

Given a matrix
⎡ ⎤
A = ⎣ ⎦
let A be a linear transformation A : R4 → R4.
A is a linear transformation that takes a vector x ∈ R4 and returns a vector y ∈ R4 such that y = Ax.
A−1 is a linear transformation that takes a vector y ∈ and returns a vector x ∈ R4 such that x = A−1y. Let
x R4
⎡1 −1 0 0 ⎤⎡x1⎤
0 1 −1 0 x2
Ax =
0 0 1 −1 x3
⎣0 0 0 1 ⎦⎣x4⎦
⎡y1⎤ ⎡x1 − x2 ⎤
y2 x2 − x3 = y3 x3 − x4
⎣y4⎦ ⎣ x4 ⎦
The above equation can be written as:
y1 = x1 − x2 y2 = x2 − x3 y3 = x3 − x4 y4 = x4
From the above equations, we can write:
x4 = y4 x3 = y3 + x4 = y3 + y4 x2 = y2 + x3 = y2 + y3 + y4 x1 = y1 + x2 = y1 + y2 + y3 + y4
Therefore,
⎡1 1 1 1⎤
0 1 1 1
x = y
0 0 1 1
⎣0 0 0 1⎦
Therefore, A−1 is a linear transformation that takes a vector y ∈ R4 and returns a vector x ∈ R4 such that x = A−1y.
⎡ ⎤
⟹ A−1 =
⎣ ⎦
Given T denotes a transformation that corresponds to rotating a vector by angle θ in the counter-clockwise direction (about the x−axis) and T1 denotes a transformation that corresponds to reflection about a given vector v ∈ R2.
As T and T1 are tranformations corresponding to rotation and reflection respectively, they are linear transformations, i.e., we have
T(x1 + x2) = T(x1) + T(x2)
T(αx3) = αT(x3)
T1 (x1 + x2) = T1 (x1) + T1 (x2) T1 (αx3) = αT1 (x3)
where x1, x2, and x3 are vectors ∈ R2 and α is a scalar ∈ R.
We have T2 (x) := T(T1 (x))
a) To show that T2 (x) is a linear transformation, we must show homogenity and additivity. Consider transformations over two vectors x1 and x2 ∈ R2.
T2 (x1 + x2) = T(T1 (x1 + x2))
⟹ T2 (x1 + x2) = T(T1 (x1) + T1 (x2)) ⟹ T2 (x1 + x2) = T((T1 (x1))) + T((T1 (x2)))
⟹ T2 (x1 + x2) = T2 (x1) + T2 (x2)
This shows that additivity is satisfied.
Now, consider a scalar α ∈ R and the vector x3 ∈ R2.
T2 (αx3) = T(T1 (αx3))
⟹ T2 (αx3) = T(αT1 (x3))
⟹ T2 (αx3) = αT(T1 (x3))
⟹ T2 (αx3) = αT2 (x3)
This shows that homogenity is satisfied.
From the two proofs above, we can conclude that T2 (x) is a linear transformation.
b) To find the matrix representation of T2, we first find the matrix representation of T and T1.
Any linear tranformation can be repeesented in the form of a matrix-vector product i.e., T(x) = Ax where A is the matrix representation of T and x is the vector representation of x.
The transformation matrix T that causes a rotation by angle θ in counter-clockwise direction (about the x−axis), is:
T x
The transformation matrix T1 that causes reflection about a given vector v ∈ R2 by some angle, say ϕ , is:
T x
Now, we find the matrix representation of T2 as follows:
T2 (x) := T(T1 (x))
[
cosθ −sinθ cos2ϕ sin2ϕ
=][ ]x sinθ cosθ sin2ϕ −cos2ϕ
[
cosθcos2ϕ − sinθsin2ϕ cosθsin2ϕ + sinθcos2ϕ
=]x sinθcos2ϕ + cosθsin2ϕ sinθsin2ϕ − cosθcos2ϕ
[
cos(θ + 2ϕ) sin(2ϕ + θ)
=]x sin(θ + 2ϕ) −cos(2ϕ + θ)