代做Math 2568 Midterm Spring 2025代写数据结构语言程序

日期：2025-02-17 05:23

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.)