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math 235 ch 3 practice

math 235 ch 3 practice - Math 235 Section 6 Chapter 3...

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Math 235 - Section 6, Chapter 3 Practice Problems Adam Gamzon 1. Find a basis of the kernel and a basis of the image of the following matrices (a) A = 1 2 3 0 - 2 1 7 0 5 (b) A = 9 3 1 1 - 3 8 - 1 - 1 0 - 1 3 1 2. What are the dimensions of the image and kernel of the matrix A = 2 4 8 4 5 1 7 9 3 ? 3. Let T : R 235 R be a linear transformation. (a) What are the possible dimensions for the image of T ? (b) What are the possible dimensions for the kernel of T ? 4. Let V and W be subspaces of R n with dim V = k and dim W = r . Let V + W := { v + w : v is in V and w is in W } . Recall that this is a subspace. Justify why dim( V + W ) k + r . (Hint: Since dim V = k , there is a basis consisting of k vectors, v 1 , . . . , v k , of V . Similarly, there is a basis of W consisting of r vectors, w 1 , . . . , w r .) 5. True or false. Justify your answer in either case. (a) The only n -dimensional subspace W of R n is R n . (b) The set W = x y z : x + y + z = 1 is a subspace of R 3 .
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