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**Example text**

Com 56 Linear Algebra Examples c-2 3. e. ⎞ ⎛ 1 0 0 ⎜ 0 1 0 ⎟ ⎟ ⎜ ⎟ Fd v = ⎜ ⎜ 0 0 1 ⎟. 8 A linear map f : C2 → C2 is deﬁned by f (v1 ) = v1 + 2v2 , f (v2 ) = iv1 + v2 , given the basis (v1 , v2 ) of C2 , 1. Find the matrix equation of f with respect to the basis (v1 , v2 ). 2. Prove that w1 = v1 + v2 and w2 = v1 − v2 form a basis of C2 . 3. Find the matrix equation of f with respect to the basis (w1 , w2 ). 1. The matrix equation is v y = Fv v (v x), where 1 2 Fv v = i 1 . 2. If w1 = v1 + v2 and w2 = v1 − v2 , then v1 = 1 1 (w1 + w2 ) and v2 = (w1 − w2 ).

2. It follows immediately that 5u1 − 3u2 = u3 , thus the dimension is at most 2. On the other hand, any two of the vectors {u 1 , u2 , u3 } are linearly independent, so the dimension is 2. Since u1 + u3 = (0, 3, 3, 3), an easy basis is 1 −u3 , (u1 + u3 ) 3 = {(1, −4, −1, 0), (0, 1, 1, 1)}, where both vectors most conveniently have a 0 as one of its coordinates. com 39 Linear Algebra Examples c-2 2. Vector Spaces 1. It follows from v1 v2 = (3, −8, 1, 4) = 3(1, −4, −1, 0) + 4(0, 1, 1, 1), = (1, −7, −4, −3) = 1 · (1, −4, −1, 0) − 3(0, 1, 1, 1), v3 v4 = (−1, 8, 5, 4) = −1 · (1, −4, −1, 0) + 4(0, 1, 1, 1), = (1, 0, 3, 4) = 1 · (1, −4, −1, 0) + 4(0, 1, 1, 1), that v1 , v2 , v3 , v4 all lie in span{u1 , u2 , u3 }, so dim span{v1 , v2 , v3 , v4 } ≤ dim span{u1 , u2 , u3 } = 2.

Find the coordinates of f (v1 ), f (v2 ) and f (v3 ) with respect to the basis d1 , d2 , d3 , d4 . Find the matrix of f with respect to the basis v1 , v2 , v3 i R3 and the basis d1 , d2 , d3 , d4 i R4 . 1. It follows from 1 0 1 0 1 −2 0 0 1 = 1 = 0, that the three vectors are linearly independent. Since the dimension of R3 is 3, we conclude that {v1 , v2 , v3 } is a basis of R3 . Then we ﬁnd ⎛ ⎞ 1 ⎜ −1 ⎟ ⎟ f (v1 ) = ⎜ ⎝ 1 ⎠, 1 and ⎛ ⎞ 1 ⎜ 0 ⎟ ⎟ f (v2 ) = ⎜ ⎝ 2 ⎠, −1 ⎛ ⎞ ⎛ ⎛ ⎞ 1 1 1 1−2+1 1 ⎜ −1 ⎟ ⎜ −1 + 0 + 1 0 1 ⎟ ⎝ −2 ⎠ = ⎜ f (v3 ) = ⎜ ⎝ 1 ⎝ 1−4+3 2 3 ⎠ 1 1 −1 −3 1+2−3 ⎞ ⎛ ⎞ 0 ⎟ ⎜ 0 ⎟ ⎟ = ⎜ ⎟.