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Math 2568

Midterm

Spring 2025

1. (20 points) Consider the vectors:

(a) Determine whether or not the set of vectors  is linearly dependent or linearly inde-pendent.

(b) Determine whether or not the set of vectors  is linearly dependent or linearly independent. (Hint: No row reduction is necessary to answer this.)

2. (20 points) Consider the linear system of equations  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 as a linear combination of the n-vectors 

(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) A

(iii) B⊤A

(iv) C ⊤C

(v) BB⊤

5. (20 points)

(a) Let  and  be solutions to the homogeneous linear system  Show that c + d 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.)