Questions and Answers #8 Matrix Transformations

Questions and Answers Sheet 8 Matrix Transformations Question #1 Matrix transformation 𝑓: 𝑅 𝟛 → 𝑅 𝟛 4 1 3 2 1 3 2 2 0     −       Establish x,y and z such that, (𝑓 ( 𝑥 𝑦𝑧 )) Do I just need to multiply the values of f for the 3 × 3 matrix? What does this mean overall? Answer: You need to solve the non-homogeneous system 4 1 3 4 2 1 3 5 2 2 0 1 x y z           − =                − Form then the augmented coefficients matrix and reduce it by rows: 1 3 1 ( ) 2 2 1 3 1 4 1 3 : 4 1 1 0 : 1/ 2 2 1 3 : 5 2 1 3 : 5 2 4 2 2 0 : 1 4 1 3 : 4 R R R R R R       −         −  − − − →         − 3 2 1 1 0 : 1/ 2 1 1 0 : 1/ 2 0 3 3 : 6 0 3 3 : 6 0 3 3 : 6 0 0 0 : 0 R R−     − −     − → −             − Well, there are infinite solutions to the above system: begin with row 3 , and remember the columns represent x,y,z from left to right, so: 𝑅2 : −3𝑦 + 3𝑧 = 6 ⇒ 𝑦 = 𝑧 − 2 𝑅1 : 𝑥 + 𝑦 = − 1 2 ⇒ 𝑥 = − 1 2 − 𝑦 = − 1 2 −𝑧 +2 = 3 2 − 𝑧 Now just choose a nice z , say 𝑧 = 0 ⇒ 𝑦 = −2 , 𝑥 = 3 2 , and one solution to your problem is (( 𝑥 𝑦𝑧 ) = ( 3/2 −20 )) as you can easily check. Question #2 Determine transformation matrix 𝑓: 𝑃2 (𝑅) → 𝑃2 (𝑅) is given by: 𝑓(𝑝(𝑥)) = 3 ⋅ 𝑝(1) − 𝑥 2 ⋅ 𝑝(0) + (𝑥 − 1) ⋅ 𝑝 ′ (1) Answer: A linear transformation can always be represented as a matrix, once you have chosen a basis for the domain and codomain. Since f is a map from a three-dimensional vector space to itself, it has a representation as a 3 × 3 matrix: 1 1 2 2 3 3 ( ) ? ? ? ( ) ? ? ? ( ) ? ? ? f p p f p p f p p             =                   where the argument to f is 𝑝(𝑥) = 𝑝1 +𝑝2𝑥 + 𝑝3𝑥 2 and the result is 𝑓(𝑝(𝑥)) = 𝑓(𝑝)1 +𝑓(𝑝)2𝑥 + 𝑓(𝑝)3𝑥 2 Now can you "probe" the '?'s by plugging in different polynomials into f? For example, if you plug in 𝑝(𝑥) = 1, which in the monomial basis is the vector 1 0 0           what is f(p), in the monomial basis? What does this tell you about the entries of the matrix? How do you find the rest of the '?'s Question #3 I have a square matrix A over 𝐶 then is its rank invariant under a congruence transformation 𝐴 → 𝑃 𝑡𝐴𝑃 ? What's the easiest way to see this? Answer: Suppose M is invertible, then the vectors 𝑣1 , … , 𝑣𝑘 are linearly independent iff the vectors M 𝑣1 , … , 𝑀𝑣𝑘 are. The same applies to 𝑀−1 , of course. It is easy to see that 𝐑𝐶 = 𝐑(𝐶𝑀) for any matrix C. It is also easy to see that 𝐑(𝑀𝐶) = 𝑀(𝐑𝐶) . In particular, the above shows that dim 𝐑 (𝑀𝐶) = dim 𝐑 𝐶 In the above case this gives 𝐑(𝑃 𝑇𝐴𝑃) = 𝑃 𝑇𝐑(𝐴𝑃) = 𝑃 𝑇𝐑𝐴, hence dim 𝐑 (𝑃 𝑇𝐴𝑃) = dim 𝐑𝐴. Question #4 Let e=(a,b,c) be a unit vector in 𝑅 𝟛 and let T be the linear transformation on 𝑅 𝟛 of rotation by 180∘ about e. Find the matrix for T with respect to the standard basis 𝑒1 = (1,0,0), 𝑒2 = (0,1,0) and 𝑒3 = (0,0,1). The rotation matrix in 𝑅 𝟛 by 180∘ is : 1 0 0 0 1 0 0 0 1   −   −       So rotating e by 180∘ gives : a b c   −   −       After that how to get the transformation matrix w.r.t the standard basis? Answer: The matrix that you gave is for a 180∘ rotation about 𝑒3 . A general way to proceed from there would be via a change-of-basis operation, which involves finding an appropriate basis and performing a couple of matrix multiplications and perhaps an inversion, to boot. Observe, however, that a 180∘ rotation about a vector e is equivalent to a reflection in the span of e, so you can save yourself quite a bit of work by using the formula for reflection of a vector v in a subspace W. This is 𝑅𝑣 = 2𝜋𝑊𝑣 − 𝑣 , where 𝜋𝑊𝑣 is the projection of v onto W. In this problem W is the span of e, so for this transformation we have 𝑇𝑣 = 2 𝑒𝑒 𝑇 𝑒 𝑇𝑒 𝑣 − 𝑣 and we know that |𝑒| = √𝑒 𝑇𝑒 = 1, so the matrix that you’re looking for is 2𝑒𝑒 𝑇 −𝐼3 . You can easily verify for yourself that Te=e and that if v is orthogonal to e, then Tv=-v , which is exactly what we want for this rotation. Question #5 Given the linear transformation: 1 0 1 0 ( ) 0 2 0 2 T M M M     = −         a) Find the matrix B of T w respect to the standard basis B of 𝑅 𝟚×𝟚 . b) Find bases of the image and kernel of B? Answer: Let a b M c d   =     from the definition of T we have: T M c   − =     So, representing M and T(M) as vectors in standard basis, T acts as: 0 0 a b b c c d         −     →             a simple inspection shows that T is represented by the matrix: 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0     −         now you can find image and kernel. Question #6 A standard matrix is given: 0 1 3 1 1 3 2 2 5 A   −   = −       − representing the linear transformation 𝐿: 𝑅 𝟛 → 𝑅 𝟛 . How to find 𝐿(2, −3, 1)? Answer: Assuming that this matrix is given with respect to the standard basis (𝑒1 , 𝑒2 , 𝑒3 ) , then the columns of the matrix are just (𝐿(𝑒1 ), 𝐿(𝑒2 ), 𝐿(𝑒3 )) , respectively. Thus, 𝐿(𝑒1 ) = (0,1,2), 𝐿(𝑒2 ) = (−1,1,2), 𝐿(𝑒3 ) = (3,−3,−5). Hence, 𝐿(2,−3,1) = 𝐿(2𝑒1 − 3𝑒2 +𝑒3 ) = 2𝐿(𝑒1 )−3𝐿(𝑒2 )+ 𝐿(𝑒3 ) = (6,−4,−7) Alternatively, for any vector 𝑣 ∈ 𝑅 𝟛 , the following is true: Lv=Av, where (𝑣 = ( 𝑥 𝑦𝑧 ) . )