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Conversely, if u + v ∈ W , er f (u) = u = 0, then ker(f ) = W . com 48 Linear Algebra Examples c-2 3. 4 Let f : R3 → R3 be the linear map which corresponds to the following matrix in the ordinary basis of R3 , ⎛ ⎞ 1 1 4 1 ⎠. F=⎝ 0 1 −1 1 −2 1. Find a basis of the range f (R3 ). 2. Prove that the vector b = (6, 2, −2) belongs to both the kernel of f and the range of f . 1. Since f (e1 ) = (1, 0, −1), f (e2 ) = (1, 1, 1) and f (e3 ) = (4, 1, −2), the range f (R3 ) is spanned by these three vectors.
Besides being a manager in the Manufacturing IT department, Kim performs triathlon at a professional level. ‘NNE Pharmaplan offers me freedom with responsibility as well as the opportunity to plan my own time. com NNE Pharmaplan is the world’s leading engineering and consultancy company focused exclusively on the pharma and biotech industries. NNE Pharmaplan is a company in the Novo Group. com 35 Linear Algebra Examples c-2 2. 17 Given in R3 the three vectors a1 = (1, 0, −1), a2 = (1, 1, 1), a3 = (1, −1, 1).
A1 + bn } are different complementary subspaces of U . com 45 Linear Algebra Examples c-2 3 3. 1 Find the matrix with respect to the ordinary basis of R3 for the linear map f of R3 into R3 , where f is mapping the vectors (2, 1, 0), (0, 0, 2) and (1, 1, 0) into (1, 4, 1), (4, 2, 2) and (1, 2, 1), respectively. Find the range of the subspace which is spanned by the vectors (1, 2, 3) and (−1, 2, 0). com 46 Linear Algebra Examples c-2 3. Linear maps hence ⎛ Ma b ⎞−1 ⎛ 1 −1 2 0 1 0 =⎝ 1 0 1 ⎠ =⎝ 0 0 2 0 −1 2 0 1 2 ⎞ ⎠ 0 and ⎛ Fd b ⎞⎛ 1 1 4 1 = ⎝ 4 2 2 ⎠⎝ 0 1 2 1 −1 −1 0 2 ⎞ ⎛ ⎞ 0 1 2 1 ⎠ = ⎝ 2 0 1 ⎠.