UMBC MATH 221: EXAM 2 QUESTIONS
AND ANSWERS. VERIFIED 2026.
What are the important postulates of Theorem 8 (Chapter 2)?
Let A be a square, nxn matrix. The following statements are all equivalent, and all true or false. -
ANS a) A is invertible
b) A is row equivalent to the nxn identity matrix
c) A has n pivot positions
d) The columns of A are linearly independent (Ax = 0 has only the trivial solution)
f) The linear transformation x → Ax is one-to-one
g) The equation Ax = b has at least one solution for each b in Rⁿ (the columns of A span Rⁿ)
l) AT is invertible
Let A and B be square matricies. If I = AB, then what is true about A and B? (2 things) - ANS 1)
A and B are both invertible
2) B = A⁻¹ and A = B⁻¹
Let T : Rⁿ → Rⁿ be a linear transformation, and let A be the standard matrix of T. When is T
invertible? And what is true of S (which is T⁻¹)? - ANS 1) If and only if A is invertible
2) S(x) = A⁻¹ x is the unique matrix satisfying S(T(x)) = x and T(S(x)) = x
@COPYRIGHT 2026/2027 ALL RIGHTS RESERVED
1
, In an LU factorization, L is a ____ triangular matrix, and U is a ____ triangular matrix. - ANS 1)
Lower Triangular
2) Upper Triangular
Describe the steps to solve Ax = b using LU factorization. - ANS 1) Change Ax = b to L(Ux) = b
2) Solve Ly = b for y (this sets Ux equal to y)
3) Solve Ux = y for x
4) The x from part 3 is your solution
What are the three properties of a subspace? (The subspace H in Rⁿ) - ANS 1) The zero vector
is in H
2) For each u and v in H, the sum u + v is in H
3) For each u in H and each scalar c, the vector cU is in H
What is the column space of A? - ANS The set of all linear combinations of the columns of A
(Col A)
What is the null space of A? - ANS The set of all solutions of the homogeneous equation Ax =
0
(Nul A or Ker A)
The null space of an mxn matrix A is a subspace of ____. - ANS Rⁿ
(Theorem 12 - Chapter 2)
What is the basis for a subspace H in Rⁿ? - ANS A linearly independent set in H that spans H
@COPYRIGHT 2026/2027 ALL RIGHTS RESERVED
2
AND ANSWERS. VERIFIED 2026.
What are the important postulates of Theorem 8 (Chapter 2)?
Let A be a square, nxn matrix. The following statements are all equivalent, and all true or false. -
ANS a) A is invertible
b) A is row equivalent to the nxn identity matrix
c) A has n pivot positions
d) The columns of A are linearly independent (Ax = 0 has only the trivial solution)
f) The linear transformation x → Ax is one-to-one
g) The equation Ax = b has at least one solution for each b in Rⁿ (the columns of A span Rⁿ)
l) AT is invertible
Let A and B be square matricies. If I = AB, then what is true about A and B? (2 things) - ANS 1)
A and B are both invertible
2) B = A⁻¹ and A = B⁻¹
Let T : Rⁿ → Rⁿ be a linear transformation, and let A be the standard matrix of T. When is T
invertible? And what is true of S (which is T⁻¹)? - ANS 1) If and only if A is invertible
2) S(x) = A⁻¹ x is the unique matrix satisfying S(T(x)) = x and T(S(x)) = x
@COPYRIGHT 2026/2027 ALL RIGHTS RESERVED
1
, In an LU factorization, L is a ____ triangular matrix, and U is a ____ triangular matrix. - ANS 1)
Lower Triangular
2) Upper Triangular
Describe the steps to solve Ax = b using LU factorization. - ANS 1) Change Ax = b to L(Ux) = b
2) Solve Ly = b for y (this sets Ux equal to y)
3) Solve Ux = y for x
4) The x from part 3 is your solution
What are the three properties of a subspace? (The subspace H in Rⁿ) - ANS 1) The zero vector
is in H
2) For each u and v in H, the sum u + v is in H
3) For each u in H and each scalar c, the vector cU is in H
What is the column space of A? - ANS The set of all linear combinations of the columns of A
(Col A)
What is the null space of A? - ANS The set of all solutions of the homogeneous equation Ax =
0
(Nul A or Ker A)
The null space of an mxn matrix A is a subspace of ____. - ANS Rⁿ
(Theorem 12 - Chapter 2)
What is the basis for a subspace H in Rⁿ? - ANS A linearly independent set in H that spans H
@COPYRIGHT 2026/2027 ALL RIGHTS RESERVED
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