235FA10-SAMPLEM2 - MTH 235 : Linear Algebra Sample problems...

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Unformatted text preview: MTH 235 : Linear Algebra Sample problems - Midterm 2 1. Prove that T is a linear transformation, and find bases for the kernel (null space) and the range of T . Is T one-to-one? Is T onto? (a) T : R 3 R 2 defined by T ( a 1 ,a 2 ,a 3 ) = ( a 1- a 2 , 2 a 3 ). (b) T : R 2 R 3 defined by T ( a 1 ,a 2 ) = ( a 1 + a 2 , , 2 a 1- a 2 ). 2. Let T : V W be a linear transformation. Prove that ker T V is a subspace. 3. Prove that a linear transformation T : V W is one-to-one if and only if ker T = { } . 4. Let V and W be finite-dimensional vector spaces and T : V W be a linear transformation. (a) Prove that if dim V < dim W then T cannot be onto. (b) Prove that if dim V > dim W then T cannot be one-to-one. 5. Let T : R 3 R 3 be the linear transformation T ( x 1 ,x 2 ,x 3 ) = (2 x 1- x 2- x 3 , 2 x 2- x 1- x 3 , 2 x 3- x 1- x 2 ) Find [ T ] B , where B is the standard ordered basis of R 3 ....
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235FA10-SAMPLEM2 - MTH 235 : Linear Algebra Sample problems...

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