[SOLVED] Math 2568 Midterm Spring 2025

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.)