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# MAT 320 Linear Algebra Final Exam, Section A Instructions: This is an open-book, open-note exam. You must work individually. Show all work relevant...

2.A projection is a linear transformation satisfying the condition T(T(v)) = T(v). Let T be a projection, and A its standard matrix. Prove A is singular - and hence, T is not invertible
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MAT 320 Linear Algebra Final Exam, Section A Instructions : This is an open-book, open-note exam. You must work individually. Show all work relevant to the solution of each problem. i.e. no credit will be given for “just the answers.” 5 Problem 1 . Let A be an n × n matrix, and W the set of eigenvectors of A with eigenvalue λ . Prove W is a subspace of R n . 5 Problem 2 . A projection is a linear transformation satisfying the condition T ( T ( v )) = T ( v ) . Let T be a projection, and A its standard matrix. Prove A is singular - and hence, T is not invertible. 10 Problem 3 . Let W be a subspace of a vector space V , and W = { w W | h w,v i = 0 v W } . a. Prove W is a subspace of V . b. For V = R 3 , with dot product as the deﬁnition of , ·i , and W = { ( 1,0,1 ) , ( 0,0,1 ) } , ﬁnd a basis for W . What is its dimension? 5 Problem 4 . Let M be the vector space of 2 × 2 matrices, and W = { A M | trace ( A ) = 0 } . Prove B , as deﬁned below, is a basis for W . B = ± ± 1 0 0 - 1 ² , ± 0 1 0 0 ² , ± 0 0 1 0 ² ² 5 Problem 5 . Deﬁne T : W W such that T ( X ) = XA - AX , where A is a ﬁxed 2 × 2 matrix and W is the subspace of M 2 × 2 with the basis B as deﬁned in Problem 4 . Prove T is a linear transformation, where A is any 2 × 2 matrix. 10 Problem 6 . With T as deﬁned in Problem 5 , and W and B as deﬁned in Problem 4 : b. Let A = ± 1 0 0 - 1 ² . Find the standard matrix for T with respect to the basis B . c. Let A = ± 1 0 0 - 1 ² . Find a basis for the kernel of T . 5 Problem 7 . Consider R , the set of all real numbers, and deﬁne + so that e x + e y = e x + y . Prove R , with this deﬁnition of addition, is a vector space. 5 Problem 8 . Deﬁne h A,B i as follows, where A and B are 2 × 2 matrices. ³± a 1 a 2 a 3 a 4 ² , ± b 1 b 2 b 3 b 4 ²´ = a 1 b 1 + a 2 b 2 + a 3 b 3 + a 4 b 4 Prove h A,B i is an inner product. 1

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