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(5) 1. 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
(5) 2. Let A =
(a) Find A−^1.
(b) Use your answer in part (a) to solve Ax = b where b =
(6) 3. Let u 1 =
x x 2
, u 2 =
x 2 x
, u 3 =
x −x
(a) For what value(s) of x will {u 1 , u 2 , u 3 } be linearly dependent? (b) For what value(s) of x will {u 1 , u 2 } be linearly dependent? (c) For what value(s) of x is Span {u 1 , u 2 } all of R^3? (d) For what value(s) of x is Span {u 1 , u 2 } a line in R^3?
(4) 4. For each of the following, find an example or explain why no such matrix is possible.
(a) A 2 × 3 matrix A so that the transformation x 7 → Ax is one-to-one. (b) A 2 × 3 matrix A where every entry is either 1 or −1 so that the transformation x 7 → Ax is not onto. (c) A matrix A such that A^2 is invertible but A is not. (d) A 2 × 2 nonzero matrix A such that A^2 = 0.
(6) 5. Let T 1 : R^2 → R^2 be the linear transformation that rotates points by π/4 radians counterclock- wise. Let T 2 : R^2 → R^2 be the linear transformation that reflects the points across the line y = −x. (a) Give the standard matrices of T 1 and T 2. (b) Give the standard matrix of T 1 ◦ T 2. (c) Let S denote the unit square in R^2 , that is S =
{[x 1 x 2
]
: 0 ≤ x 1 ≤ 1 and 0 ≤ x 2 ≤ 1
Draw pictures of S and (T 1 ◦ T 2 )(S).
(5) 6. Let A =
[ 1 − 2
]
(a) For what value(s) of k is
[ 3
k
]
in Col A?
(b) For what value(s) of k is
[ 3
k
]
in Nul A? (c) Give a basis for Nul A^2. (d) Is Nul A= Nul A^2? Justify your answer.
(3) 7. Suppose A and B are n × n matrices where A has linearly independent columns and B is invertible. (a) Simplify (BAB−^1 )^2. (b) Simplify (BAB−^1 )−^1. (c) Does BAB−^1 have linearly independent columns? Justify your answer.
(7) 8. Fill in the blanks. The missing word is 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 x ∈ Col A and y ∈ Col A then x + y be in Col A. (e) Suppose A is a 5 × 7 matrix then Row A and Col A have the same dimension. (f) Suppose A is a 5 × 7 matrix then Nul A be 3 dimensional. (g) Suppose A is a 5 × 7 matrix of rank 4, then Nul AT^ be 3 dimensional.
(6) 9. Let A =
(a) Find lower triangular matrix L and upper triangular matrix U so that A = LU. (b) Do the same for AT^. (Hint: No additional computation is required.) (c) What is det A?
(d) Find an elementary matrix E such that EA =
(6) 10. Let U be an n × n matrix which is partitioned as U =
[ 0 I
A B
]
(a) Assume A is invertible. Write U −^1 as a partitioned matrix.
(b) Use part (a) to find the inverse of M where M =
(7) 11. Let A =
,^ b^ =
and^ x^ =
x 1 x 2 x 3 x 4
(4) 18. Suppose A = [a 1 a 2 · · · an] is an n × n matrix such that ∥Ax∥ = ∥x∥ for all x ∈ Rn.
(a) Show that each column of A is a unit vector. (Hint: consider ai = Aei.) (b) Show that any two different columns of A, ai and aj , are orthogonal. (Hint: Consider the result in part (a) and ∥ai + aj ∥^2 .) (c) Show that AT^ A = In.
(4) 19. A matrix X is called a weak generalized inverse of A if
AXA = A
(a) For what value of k is
k k k k k k
(^) a weak generalized inverse of
[ 1 1 1
]
For parts (b) and (c), suppose that X is a weak generalized inverse of m × n matrix A (so you know that AXA = A even though A is not necessarily invertible). (b) Show that if y is any vector in Rn, then (I − XA)y is in Nul A. (c) Show that if the system Ax = b is consistent then Xb will be a solution to this system.
