Test 1 Practice Questions (First set) (1) Consider the following linear ...

Test 1 Practice Questions (First set)
(1) Consider the following linear system and answer the questions.
⎧
⎨
⎩
x + y − z = −2
3x − 5y + 13z = 18
x − 2y + 5z = k
(a) Write down the augmented matrix.
(b) For which k does this system have one solution?
(c) Is the coefficient matrix A invertible? Explain your answer.
(d) Let A be the coefficient matrix of this linear system and let T be the linear
transformation defined by A. Find the kernel and image of T .
(2) Consider the following linear system and answer the questions.
⎧
⎨
⎩
x + 2y + 3z = 4
x + ky + 4z = 6
x + 2y + (k + 2)z = 6
(a) For which k does this system have a unique solution?
(b) When is there no solution?
(c) When are there infinitely many solutions?
(3) Find the polynomial of degree 2 whose graph goes through the points (1, 1), (2, 3)
and (3, 13).
(4) Find all vectors in R3 that are perpendicular to the following two vectors:
⎡
v1 = ⎣ 1
⎤
⎡
3 ⎦ , and v1 = ⎣
0
1
⎤
1 ⎦ .
1
(5) The cross product of two vectors in R3 is given by
⎡ ⎤ ⎡ ⎤ ⎡
⎣ a1
a2
a3
⎦ ×
⎣ b1
b2
b3
⎦ =
⎣ a2b3 − a3b2
a3b1 − a1b3
a1b2 − a2b1
Now fix an arbitrary vector v in R 3 . Is the transformation T (x) = v × x from R 3 to
R 3 linear? If so, find its matrix in terms of the entries of v.
(6) Suppose a line L in R3 contains the vector
⎡ ⎤
u =
⎣ a1
a2
Find the matrix A of the linear transformation T (x) = proj L(x).
1
a3
⎦ .
⎤
⎦ .

(7) For
A =
⎡
⎢
⎣
0 1 2
0 2 4
0 3 6
1 4 8
(a) Find a vector b in R 4 such that the system Ax = b is inconsistent;
(b) Find a vector c in R 4 such that the system Ax = c is consistent.
1 10
(8) Let A =
. Find a scalar λ such that A − λI2 fails to be invertible. For
−3 12
each scalar λ you find, find a nonzero vector x such that Ax = λx.
(9) Answer the following questions
(a) Describe the kernel of the matrix ⎣
⎡
⎤
⎥
⎦ ,
1 1 1
1 2 3
1 3 5
(b) Describe the image of the linear transformation T (x) = Ax for A = ⎣
(10) For the matrix
⎡
A = ⎣
0 1 0
0 0 1
0 0 0
describe the images and kernels of the matrices A, A 2 , A 3 .
⎤
⎦;
⎡
1 1 1
1 1 1
1 1 1
(11) Consider a n × p matrix A and a p × m matrix B such that ker(A) = {0} and
ker(B) = {0}. Find ker(AB).
(12) For two invertible n × n matrices A and B, determine which of the formulas below
are necessarily true.
(a) (A + B) 2 = A 2 + 2AB + B 2
(b) A 2 is invertible and (A 2 ) −1 = (A −1 ) 2
(c) A + B is invertible, and (A + B) −1 = A −1 + B −1
(d) (A + B)(A − B) = A 2 − B 2
(e) ABB −1 A −1 = In
(f) ABA −1 = B
(g) (ABA −1 ) 3 = AB 3 A −1
(h) (In + A)(In + A −1 ) = 2In + A + A −1
(i) A −1 B is invertible, and (A −1 B) −1 = B −1 A.
2
⎤
⎦ ,
⎤
⎦.