Linear transformation, transformation matrices

A32: Matrices a) Name a base (with explanation!) of ℝ” $ %. b) Given are these matrices: Calculate all possible matrix products between the matrices A, B, C (each matrix should appear only once in the respective product). c) Let A = . Calculate An for n ∈ â„•. Note: Calculate A1, A2, A3, etc. first. A33: Compositions of linear transformation The linear transformations 𝐺 ℝ4 → ℝ3,𝐺 ℝ3 → ℝ3 𝐵𝐠ℎ: ℝ3 → ℝ4 are defined by: Calculate with the help of the transformation matrix(regarding the standard basis) following compositions and specify the function rule in Cartesian coordinates: A34: Linear transformation, transformation matrices, Nullspace and image The linear transformation 𝐺 ℝ4 → ℝ4 is defined by: a) Show that g is well-defined, i. verify that the vectors (1,1,0,0)T, (1,0,1,0)T, (0,1,0,1)T, (0,0,1,2)T are linearly independent and thus form a basis in ℝ3 and specify the transformation matrix of g (regarding to that basis). b) Name the transformation matrix of M of g regarding the standard basis. Note: First determine the representation of the basis vectors 𝐵 , i = 1, 2, 3, 4 in the basis c) Calculate the nullspace of g and image of g.

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