lina-04-12-09-p1 - Linear Algebra & Geometry: Sheet 9...

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Unformatted text preview: Linear Algebra & Geometry: Sheet 9 Set on Friday, Dec 4: Questions 1, 2, 3 and 4 1. Consider the following matrices A= ￿ ￿ 00 12 B= ￿ 1 −1 ￿ ￿ C= 4 2 ￿ 0 −1 D = 2 0 31 E= ￿ ￿ 2 01 −1 3 0 and determine which of them can be multiplied, and in which order, and compute the products. 2. Let us associate with each complex number z = x + iy ∈ C the following matrix ￿ ￿ x −y A(z ) := . yx Show that (ii) for all z, z ￿ ∈ C we have A(z )A(z ￿ ) = A(zz ￿ ). (i) for all z, z ￿ ∈ C we have A(z ) + A(z ￿ ) = A(z + z ￿ ). (iii) compute the inverse matrix of A(z ), i.e., find a 2 × 2 matrix Bz such that Bz A(z ) = I . 3. Let T : Rn → Rm be a linear map and MT = (tij ) the corresponding m × n matrix. Let mj , j = 1, 2, · · · , n, be the column vectors of MT , i.e., mj = (t1j , t2j , · · · , tmj ). Show that (i) T (ej ) = mj and T (x) = x1 m1 + x2 m2 + · · · + xn mn (iii) the image of T is spanned by the column vectors m1 , m2 , · · · , mj of MT , i.e., Im T = span{m1 , m2 , · · · , mn }. 4. Consider the map T : R2 → R2 given by T (x, y ) = (2x − y, 8x − 4y ) . (i) Determine Im T . For which of the following elements b of R2 does there exist a solution x to T (x) = b ￿￿ ￿￿ ￿ ￿ 0 2 −1 1 8 7 (ii) Determine ker T . Are the column vectors of MT linearly independent? (ii) the column vectors m1 , m2 , · · · , mn of MT are linearly independent if, and only if, ker T = {0}. 1 ...
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This note was uploaded on 02/25/2010 for the course MATHEMATIC 11007 taught by Professor Spyros during the Spring '10 term at Bristol Community College.

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