Linear Algebra with Applications (3rd Edition)

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Unformatted text preview: Math 253, Fall SHINE. Exam #2 Exam # 2 . . . . _. 3 2 _. , 1. Find the maths of the linear transformation T(I)= 4 5 I “nth respect to the _ i] —1 hast , . 2] l 1—). (i 2 2. For which values of the constant A is the matrix A: 5 —)t s invertible“? 4 D 3—). 3. True or False? a. If»! isaninvertible nxn matrix, then the equation [A4]; = (AT) 1 musthoiri. 116 b. There exists an orthogonal 3X3 matrix whose first column is I] —U.E c. All linear transfonnations from P; to lit“ are isomorphisms. d. The rank of the linear transformation if":ij : f’ (the derivative] from P: r to E, is n, for all positive integers n. i D Ii! 15 11' 3 2 fl '1 fl 1 s isinvertihle for all real numbers I. B Iii ? I1 e. The matrix at: LA l-JI WEI D‘- H 2-: Ln.‘ Lu- 3 4. Let Vb: the set ofali polynomials f[:r} in P2 such that ff[x]rit=3f[l].Weare fl told that V is a subspace of P2 . Find a basis of V and determine the dimension of V. 5. Let V be the space oi" a]! symmetric 2x2 matrices with the basis 2% consisting of 1 o] o 1 o 0" {J D 1 fl .Consider the linear transformation L[A)=STAS from V ,and to V_where S=ifl 1i. Math 253. Fall EDGE. Exam #1 Solution 5.a. ab {lease} a e a e —'—|* [125: at) 2:0 a+4b+4e l L a {} s t} r: —-—r a+4b+4e nee s=e e e 144 at: h. E] is a basis of the image efB, so that is a basis of the image ef L. l a 4 e. —l , t] is a basis of the kernel ef B (eensider the Kyle numbers), se that D ~r 1 4 —I 4 e . . , is a basis ef the kernel at L. —I G U —1 cl. Ne, L isn't an isemerphism, sinee ketiL} eensists ef' mere than just the zeta matri: ...
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