Math NYC Practice Test 1
(Note: Answers are typed at the end of the questions on Page2)
1. (a) Solve the following system. (b) Write your answer in parametric-vector form. (c) Using only part
(b) determine the solution of the corresponding homogeneous system. (d) Give 3 nontrivial solutions
for this homogeneous system. (e) Give a dependency equation for the columns of the coefficient
matrix, A, of the system.
3x1 + 6x2 − 5x3 + 7x4 = 7
x1 + 2x2 − 2x3 + 3x4 = 2
2x1 + 4x2 − 3x3 + 4x4 = 5
5x1 + 10x2 − 8x3 + 11x4 = 12
2. For what value(s) of k, will the following system have a) no solution b) a unique solution or c)
infinitely many solutions?
x + ky = 4
(k − 1)x + 2y = 4
3. Find the interpolating quadratic polynomial p(x) = a0 + a1 x + a2 x2 that passes through the points
(1, 2), (−1, 6), and (2, 3)
4. Does u = (8, −3, −6) belong to Span{(1, 0, 3), (−2, 1, 4)}? Why or why not?
1 1 1 1
2 0 4 1
5. Determine if the set 1 , 1 , 1 , 1 is linearly independent. If not give a depen-
1 −1 3 0
dency equation.
1 2 a
6. Give conditions on a, b and c so that 1 , −1 , b is linearly independent.
1 1 c
1 3 0 4 2 0 0
0 0 1 2 0 0 0
7. Suppose A is a 4 × 7 matrix the RREF of which is R = 0 0 0 0 0 1 0
0 0 0 0 0 0 1
(a) Determine, without doing any calculations, whether Ax = 0 has a unique solution, infinitely
many solutions, or no solution. Justify your answer.
(b) Are columns of A linearly independent? Justify.
(c) Do columns of A span R4 ? Justify.
(d) Consider the linear transformation T (x) = Ax. Is T one-to-one? Is T onto? What is range of T ?
8. Determine if each of the following statements is TRUE or FALSE? Explain your answer in each case.
(a) If {v1 , v2 , v3 , v4 } is a linearly independent set in R4 , then so is {v1 , v2 , v3 }.
(b) If a matrix A has a pivot position in every column, then its columns are linearly independent.
(c) The equation x = p + tv, t ∈ R describes a line through v and parallel to p.
(d) If u and v are any vectors in R3 , then Span{u, v} is a plane through origin.
(e) If A is a 3 × 5 matrix and T (x) = Ax. Then domain of T is R3 .
(f) Suppose {v1 , v2 } is a linearly independent set in R3 and w ∈ Span{v1 , v2 }. Then {v1 , v2 , w} is
linearly independent.
1
9. Let T : R2 → R4 be defined by T (x1 , x2 ) = (2x2 − 3x1 , x1 − 4x2 , 0, x2 ). (a) Find the standard matrix
of T . (b) Is T one-to-one? Is it onto? Justify your answers. (c) What is range of T geometrically?
(d) Is (1, 2, 5, 6) in range of T ?
10. Let {v1 , v2 , v3 , v4 } be a linearly independent set in Rn . Suppose that v = c1 v1 + c2 v2 + c3 v3 + c4 v4
with c1 6= 0. Prove that {v, v2 , v3 , v4 } is also linearly independent.
Suggested Additional Problems
Textbook (2nd or 4th custom made edition)
1.3: 25, 26
1.4: 17–20
1.5: 17, 19, 21, 27, 34, 36, 40
1.7: 35–40
1.8: 25–28, 31
1.9: 1–14
Supplementary Exercises for Chapter 1: 1, 5–23
Answers
x1 4 −2 1
x2 0 1
+ t 0
1. x =
x3 = 1 + s 0
2 s, t ∈ R
x4 0 0 1
x1 −2 1
x2 1 0
xh =
x3 = s 0 + t
s, t ∈ R
2
x4 0 1
Three nontrivial solutions for the homogeneous system are, e.g., (1, 0, 2, 1), (−2, 1, 0, 0), (−1, 1, 2, 1)
(e) using the third nontrivial solution, one dependency equation is −a1 + a2 + 2a3 + a4 = 0 (many
answers possible)
2. No solution if k = −1; a unique solution if k 6= −1 and k 6= 2; infinitely many solutions if k = 2
3. p(x) = 3 − 2x + x2
4. Yes: (8, −3, −6) = 2(1, 0, 3) − 3(−2, 1, 4)
5. Linearly dependent: Two dependency equations are (many answers possible):
2a4 = a1 + a2
a3 = 2a1 − a2
6. 2a + b − 3c 6= 0
7. (a) infinitely many solutions; (b) No; (c) Yes; (d) T is not 1-1, yet it is onto (Justify by looking at
the pivot positions). The range of T is R4 .
8. (a) True; (b) True; (c) False; (d) False; (e) False; (f) False
−3 2
1 −4
9. (a) A = 0
(b) This transformation is 1-1 but not onto (Note columns of A are linearly
0
0 1
independent; however, there is a row of zeroes in A). (c) The range of T is a plane in R4 . (d) No!
10. Hint: use the equation dv + d2 v2 + d3 v3 + d4 v4 = 0 and show all the coefficients must be zero.
