MATH70 final Math70-FinalReview

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Published on 31 Jan 2019
School
Tufts University
Department
Mathematics
Course
MATH-0070
Professor
MATH 46, REVIEW FOR FINAL
1. (a) Let AMm×n,A= [aij ]. Deﬁne the following terms: cofactor of aij , det A,eigenvalue
of A,eigenvector associated to an eigenvalue of A, eigenspace associated to eigenvalue of
A,Ais diagonalizable.
(b) For a function T:VVdeﬁne the following terms: Tis a linear transformation,Tis
diagonalizable.
(c) Describe the least squares method to solve Ax=b. When might you use it to solve
Ax=b?
(d) What deﬁnitions do you think will be on the test?
2. Let AMm×n. Assume for all xRnthat Ax=0. Prove that Ais the zero matrix.
3. Solve the following linear system:
2x+y2z= 10
3x+ 2y+ 2z= 1
5x+ 4y+ 3z= 4
(a) by row reduction.
(b) by Cramer’s Rule. [NOT COVERED IN 2018]
4. Let Abe an m×nmatrix and let TA:RnRmbe deﬁned by TA(x) = Ax. Are the following
statements true or false. If true give a proof, if false explain why.
(a) dim Nul An.
(b) rank Am.
(c) If n > m then the linear transformation TAcannot be one-to-one.
(d) If n < m then TAcannot be onto.
(e) If TAis one-to-one and m=nthen TAmust be onto.
5. Let Abe an n×nmatrix satisfying A3=In, where Inis the n×nidentity matrix. Show that
det A= 1.
6. For the following problems, prove the statement or give a speciﬁc counterexample:
(a) W={pP2
(p(3))2+p(3) = 0}is a subspace of P3.
(b) W={(x, y)R2
x+y= 0}is a subspace of R2.
(c) Let Vbe a vector space and let f1:VRand f2:VRbe linear transformations.
Dene T:VR2dened by T(v) = (f1(v), f2(v)). Tis linear.
7. Let S=2 1
0 1 ,1 0
1 1 ,8 2
4 6 .
(a) Decide whether Sis independent.
(b) Let W= span S. Use the result of (a) to nd a basis of Wthat is a subset of S. Find
dim W.
(c) Determine whether 5 1
1 9 W= span S.
8. If you are given a square upper triangular matrix, how would you tell at a glance whether or
not it is invertible? Explain your answer using determinants. How would you tell at a glance
the eigenvalues of the matrix and their multiplicities?
9. Let Vand Wbe vector spaces and let T:VWbe a linear transformation.
(a) Let S={v1,v2,v3,...,v}be a set of vectors that spans V. Prove that the set
{T(v1), T (v2), T (v3), . . . , T (v)}spans range Tin W.
(b) Assume Tis one-to-one, and assume Sis a basis of V. Prove the set
{T(v1), T (v2), T (v3), . . . , T (v)}is independent.
(c) Show that L:P3R3be dened by L(p) = (p(0), p(1), p(3)). Prove Lis a linear
transformation.
(d) Use the result of (a) and the basis B={1, t, t2, t3}of P3to nd a spanning set of range L.
Find a subset of this spanning set that is a basis of range L.
1
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