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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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