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- (a) Solve the system: x 1 + x 2 − x 3 − 2 x 4 + x 5 = 1 2 x 1 + x 2 + x 3 + 2x 4 − x 5 = 2 x 1 + 2x 2 − 4 x 3 − 8 x 4 + 5x 5 = 1 x 2 − 3 x 3 − 6 x 4 + 3x 5 = 0
(b) Write the zero vector in R^4 as a nontrivial linear combination of the columns of A, where A is the coefficient matrix for the system of equations in part a) of this question.
- Let A =
(a) Find A−^1.
(b) Use your answer in part (a) to solve Ax = b where b =
- Let T : R^2 → R^3 be defined by T
([
x y
])
x + 1 2 y x − 1
(a) Is T linear? Justify. (b) Is T one-to-one? Justify. (c) Is T onto? Justify.
(d) Sketch the line
[
]
[
]
then find its image under T.
- Give an example of each of the following. If no such example is possible, explain why.
(a) A 2 × 3 matrix A such that the transformation x 7 → Ax is one-to-one. (b) A 2 × 3 matrix A where every entry is either 1 or −1 such that the tranformation x 7 → Ax is NOT onto. (c) A matrix A such that A^2 is invertible but A is not. (d) A nonzero matrix A such that A^2 = 0.
- Let A and B be n × n matrices where B is invertible and A has linearly independent columns.
(a) Simplify (BAB−^1 )^2. (b) Simplify (BAB−^1 )−^1. (c) Does BAB−^1 have linearly independent columns? Justify your answer.
- Let A =
[
]
(a) For which value(s) of k is
[
k
]
in Col(A)?
(b) For which value(s) of k is
[
k
]
in Nul(A)?
(c) Give a basis for Nul(A^2 ). (d) Is Nul(A) =Nul(A^2 )? Justify your answer.
- Fill in each blank with the missing word. In each case, the missing word is either, must, might or cannot.
(a) If y ∈ Col(A) then Ax = y be inconsistent. (b) If y ∈ Col(A) then y be in Nul(A). (c) If y ∈ Col(A) then y be in Row (AT^ ). (d) If y ∈ Col(A) and x ∈ Col(A) then x + y be in Col(A). (e) If A is a 5 × 7 matrix then Row(A) and Col(A) have the same dimension. (f) If A is a 5 × 7 matrix then Nul(A) be three-dimensional. (g) If A is a 5 × 7 matrix of rank 4, then Nul(AT^ ) be three-dimensional. (h) If u and v are linearly independent then Projuv and Projvu be equal.
- Let W be an n × n matrix that is partitioned as W =
[
0 I
A B
]
, where the matrix A is known to be invertible. (a) Write W −^1 as a partitioned matrix.
(b) Use part (a) to find M −^1 where M =
- Let A =
(a) Find a lower triangular matrix L and an upper triangular matrix U such that A = LU. (b) Do the same for AT^. (Hint: No additional computation is required.)
(c) Find an elementary matrix E such that EA =
- Let A =
, let^ b^ =
and let^ x^ =
x 1 x 2 x 3 x 4
(a) Find det(A). (b) Use Cramer’s Rule to solve Ax = b for x 4 ONLY. (c) What is det(A−^1 AT^ )? (d) What is det(A · adj(A))?
(e) Find the determinant of B =
, noting that^ B^ is obtained from^ A^ by performing
exactly two elementary row operations.
- Let u 1 =
x x 2
, u 2 =
x 2 x
, u 3 =
x −x
(a) For which value(s) of x will {u 1 , u 2 } be linearly dependent? (b) For which value(s) of x will Span{u 1 , u 2 } be all of R^3? (c) For which value(s) of x is Span{u 1 , u 2 } a line in R^3?
Solutions 1. a) x 1 = − 2 x 3 − 4 x 4 + 1, x 2 = 3x 3 + 6x 4 , x 3 is free, x 4 is free, x 5 = 0 b) 0 =
− 6 a 1 + 9a 2 + a 3 + a 4 where ai is the ith^ column of A. 2. a) A−^1 =
(^) b)
x =
(^) 3.a) No, since T ( 0 ) 6 = 0 b) Yes; prove that if T (v 1 ) = T (v 2 ) then v 1 = v 2 c) No, since
for example the zero vector is not in Range(T ) d) The image of the line is (3, 2 , 1) + t(− 1 , 4 , −1) 4. a)
Impossible b)
[
]
c) Impossible d)
[
]
- a) BA^2 B−^1 b)B(BA)−^1 c) Yes 6.
a) k = 6 b) k = 3/2 c) B =
{[
]}
d) They are equal. 7. a) cannot b) might c) must d) must e) must
f) might g) cannot h) cannot 8. a) W −^1 =
[
−A−^1 B A−^1
I 0
]
b) M −^1 =
9.a) A = LU =
(^) b) AT^ = (LU )T^ = U T^ LT^ c) E =
(^) 10. a)
18 b) − 7 /3 c) 1 d) (18)^4 e) − 18 11. a) x = 2 b) Impossible c) x = 2 d) x = ± 2 12. a) Yes b) Yes c)
No d) No 13. a) No b) (3, 1 , 1)+t(1, − 4 , 2) c)
d) The angle is
π 2
−cos−^1 (− 2 /
- e) No. It doesn’t
pass through the origin. 14. a) (0, 1 , −1) + s(2, 2 , 3) + t(2, 1 , −3) b)
342 + 31^2 + 33^2
c) 141 (− 6 , 11 , −5)
- B = {x^2 − x + 1} and dim(V )=1 16. a) The vectors in B are linearly independent and span H. b) Show that {T (v 1 ), T (v 2 ), ...., T (vn)} is a basis for T (H). 17. a) ||ai|| = ||Aei|| = ||ei|| = 1 b) ||a 1 +aj ||^2 = ||Aei +Aej ||^2 = ||A(ei +ej )||^2 = ||ei +ej ||^2 = ||ei||^2 +||ej ||^2 = ||Aei||^2 +||Aej ||^2 = ||ai||^2 +||aj ||^2 Conclusion: ai and aj are orthogonal. c) The entries along the diagonal are of the form ai · ai = ||ai||^2 = 1 and the entries off the diagonal are of the form ai · aj = 0 where i 6 = j since ai and aj are orthogonal. d) Any rotation matrix would work.
