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# PSU MATH 220 - Spring 2022 Final Exam Review

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Math 220 Spring 2022Final Exam Review1. The matrices given below are in echelon form with ∗ representing a nonzero entry. Eachaugmented matrix represents a linear system. Determine if the system is consistent.If it is consistent, determine if the solution is unique.(a)∗ ∗ ∗ ∗0 ∗ ∗ ∗0 0 ∗ 0(b)0 ∗ ∗ ∗ ∗0 0 ∗ ∗ ∗0 0 0 0 ∗(c)∗ ∗ ∗0 0 00 0 02. Consider the systemx1+ hx2= 24x1+ 8x2= kChoose h and k so that the system has(a) no solution(b) a unique solution(c) many solutions.3. Give a geometric description of Span{v1, v2, v3}.(a) v1=82−6v2=123−9v3=431(b) v1=82−6v2=123−9v3=41−3(c) v1=100v2=010v3=00114. Solve the following vector equations, put your answers in parametric vector form, thencompare your work and answers.(a)3 2 7 01 4 −1 0(b)3 2 7 141 4 −1 8(c)3 2 7 −31 4 −1 95. Determine if each set is linearly independent. If the set is linearly dependent, give anexample of how the third vector can be written as a linear combination of the othervectors. Then give an example of how the second vector can be written as a linearcombination of the other vectors.(a)03−11,−8−75−3,54−42(b)123,10−1,−1496. Suppose T : R2→ R2is a linear transformation such thatT52=21and T13=−13.Find T−109.7. Assume that T is a linear transformation. Find the standard matrix of T .(a) T : R2→ R2rotates points about the origin through3π2radians counter-clockwise.(b) T : R2→ R2reflects points across the x-axis, then reflects across the line y = x.(c) T : R3→ R3projects onto the xz-plane.(d) T : R3→ R2defined by T (x1, x2, x3) = (x1− 5x2+ 4x3, x2− 6x3).8. Is the transformation in 7 (d) one-to-one? Is it onto?9. Find the inverse of each matrix or determine the matrix is not invertible.(a)8 35 2(b)1 23 62(c)1 0 −2−3 1 42 −3 4(d)1 −2 14 −7 3−2 6 −410. Determine which sets are a basis for R2.(a)5−2,10−3(b)−46,2−3(c)5−2,10−3,−11(d)10,01(e)1011. A matrix A and its echelon form are given below. Find a basis for Col(A) and Nul(A).(a) A =4 5 9 −26 5 1 123 4 8 −3⇝1 2 6 −50 1 5 −60 0 0 0(b) A =1 4 8 −3 −7−1 2 7 3 4−2 2 9 5 53 6 9 −5 −2⇝1 4 8 0 50 2 5 0 −10 0 0 1 40 0 0 0 012. Which of the following subsets of R2are subspaces? Give a geometric interpretationwhenever possible.(a) H =xy|x ≥ 0(b) H =x−2|x ∈ R(c) H =xx|x ∈ R(d) H =xy|y ≥ −x313. Find a basis for the subspace spanned by the vectors1−32−4,−39−612,2−142,−45−37.What is the dimension of the subspace?14. The vector x is in a subspace H with a basis B = {b1, b2}. Find the B-coordinatevector of x.(a) b1=1−4, b2=−27, x =−37(b) b1=15−3, b2=−3−75, x =410−715. Compute the determinant by cofactor expansion.(a)3 0 42 3 20 5 −1(b)2 −2 33 1 21 3 −1(c)4 0 0 51 7 2 −53 0 0 08 3 1 7(d)3 5 −6 40 −2 3 −30 0 1 50 0 0 3(e)4 0 −7 3 −50 0 2 0 07 3 −6 4 −85 0 5 2 −30 0 9 −1 216. Find the determinant by row reduction to echelon form.(a)1 5 −4−1 −4 5−2 −8 7(b)1 3 0 2−2 −5 7 43 5 2 11 −1 2 −3417. Let A and B be 4 ×4 matrices, with det A = −3 and det B = −1. Compute(a) det(AB)(b) det(B5)(c) det(2A)(d) det(ATBA)(e) det(B−1AB)18. Suppose thata b cd e fg h i= 7.Find each of the following determinants.(a)a b cd e f3g 3h 3i(b)a b cd + 3g e + 3h f + 3ig h i(c)a b c3d + a 3e + b 3f + cg h i(d)3a 3b 3c3d 3e 3f3g 3h 3i19. Is4−31an eigenvector of3 7 9−4 −5 12 4 4? If so, find the eigenvalue.20. Is1−21an eigenvector of2 6 73 2 75 6 4? If so, find the eigenvalue.21. Let A =7/2 −1/2 0−1/2 7/2 00 0 1. Which of the following vectors are eigenvectors of A?(a) (1,0,1) (b) (0,1,1) (c) (-1,1,0) (d) (1,1,-1)22. Find the characteristic equation of A =−1 0 1−3 4 11 0 2.523. Let A =2 5 30 6 00 4 3. Find the eigenvalues of A2.24. Find a basis for the eigenspace corresponding to each listed eigenvalue.(a) A =10 −94 −2, λ = 4(b) A =3 −1 3−1 3 36 6 2, λ = −4(c) A =4 2 3−1 1 −32 4 9, λ = 325. Let A =5 02 1. Find all eigenvalues of A. Then find a basis for the eigenspacecorresponding to each eigenvalue.26. Find h in the matrix A below so that the eigenspace of A corresponding to λ = 4 istwo-dimensional.4 −3 9 70 1 h 00 0 4 50 0 0 −327. Diagonalize the following matrices, if possible.(a)1 06 −1(b)5 10 5(c)3 −11 5(d)2 34 128. Compute Akwhere A =4 −32 −1.29. A matrix has eigenvalues λ1= 2 and λ2= −1 and corresponding eigenvectors v1=3−1and v2=−21. Find A.630. Which of the following matrices is not diagonalizable?(a)3 2 50 4 70 0 9(b)2 0 05 6 04 9 0(c)1 2 00 1 00 0 1(d)1 2 02 1 00 0 131. Find the length of the vector −3v if v = (2, 3, −2, 1).32. Let y = (1, 2, 6) and u = (1, 1, 1). Compute the distance from y to the line through uand the origin.33. Find the orthogonal complement of each subspace.(a) W =span010,001(b) W =col1 4 52 5 73 3 634. Let W be a subspace

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