Math 2568 Midterm Spring 2025
1. (20 points) Consider the vectors:
(a) Determine whether or not the set of vectors {⃗v1, ⃗v2, ⃗v3} is linearly dependent or linearly inde
pendent.
(b) Determine whether or not the set of vectors { ⃗w1, ⃗w2, ⃗w3, ⃗w4} is linearly dependent or linearly
independent. (Hint: No row reduction is necessary to answer this.)
2. (20 points) Consider the linear system of equations A⃗x = ⃗b with augmented matrix
In
(a)–(c), a matrix
in echelon form. which is row equivalent to the augmented matrix is given.
In each case, determine whether the original system:
(i) is inconsistent
(ii) has a unique solution
(iii) has infinitely many solutions; in this case, find the general solution.
3. (20 points)
Find a number b so that the matrix
is singular.
4. (20 points) Let
be an m × n matrix,
be an n × p matrix,
be an p × q matrix,
be an n-vector, and
be a p-vector.
(a) Express B ⃗w as a linear combination of the n-vectors B⃗
1, . . . , B⃗
p.
(b) Suppose m, n, p, and q are all different integers. Determine which of the following products
are defined and find their dimensions:
(i) B⊤C
(ii) A⃗v
(iii) B⊤A
(iv) C
⊤C
(v) BB⊤
5. (20 points)
(a) Let ⃗v and ⃗w be solutions to the homogeneous linear system A⃗x = ⃗0. Show that c⃗v + d ⃗w is also
a solution to this system.
(b) Let A and B be two n × n matrices. Show that if B is singular, then AB must be singular.
(Hint: Consider the homogeneous system definition of singularity.)
