SAMPLE PROBLEMS FOR MIDTERM I, MATH265

1-4.Consider the system of linear equations

(a) Write down the augmented matrix associated to this system of linear equations

(b) Apply the Gauss-Jordan elimination to reducing the augmented matrix to the reduced row echelon form.

(c) Describe the set of solutions using (b).

1.

\begin{aligned}x_{1} + 2x_{2} + 3x_{3} & = & 9 \\ −x_{1} + 3x_{2} + 2x_{3} & = & 1 \\ 2x_{1} − x_{2} + x_{3} & = & 8 \end{aligned}

2.

\begin{aligned}x_{1} + 2x_{2} − x_{3} + x_{4} & = 1 \\ 2x_{1} + 4x_{2} − 2x_{3} + 2x_{4} & = 2 \\ 5x_{1} + 10x_{2} − 5x_{3} + 5x_{4} & = 5 \end{aligned}

3.

x_{1}  −x_{3} +3x_{4} = 0
2x_{1} +x_{2} −2x_{3} +5x_{4} = 3
x_{1} −x_{2}+ −x_{3} +4x_{4} = −3
6x_{1} +3x_{2} −3x_{3} +12x_{4} = 10

4.

x_{1}+ x_{2} + −x_{3} +2x_{4} = 0
−x_{1}+ x_{2} + 4x_{3} −3x_{4} = 0
x_{2} + 5x_{3} +x_{4} = 0

5. Determine whether the following matrices are row equivalent:

A = [ \begin{matrix}2 & −1 & 0 & 1 & 0 \\ 0 & 1 & 1 & 0 & 1 \\ 1 & −1 & 0 & 1 & 1 \end{matrix} ]

B = [ \begin{matrix}1 & 0 & −1 & 0 & 1 \\ 1 & 0 & −2 & 1 & 1 \\ 1 & 1 & 1 & −1 & 1 \end{matrix} ]

Hint: Find their rref form.

6-8. For which values a (and b ) does the system of linear equations [a] have infinitely many solutions; [b] have no solution; [c] have a single solution

6.

\begin{aligned}x_{1} & −x_{2} & −x_{3} & = & 1 \\ 2x_{1} & 2x_{2} & +x_{3} & = & a \\ 5x_{1} & +3x_{2}+ & x_{3} & = & −a^{2} \end{aligned}

7.

x_{1} −x_{2} −x_{3} = 1
x_{2}  +x_{3} = a
x_{1} +2x_{2}+ ax_{3} = a

8.

\begin{aligned}x_{1} & +2x_{2} & +bx_{3} & = & a − 1 \\ x_{1} & +3x_{2} & +(a + b)x_{3} & = & a \\ 2x_{1} & +4x_{2}− & (a + b)x_{3} & = & 2a − 1 \end{aligned}

9.Let

A = [ \begin{matrix}0 & 1 & 1 \\ 1 & 0 & 2 \\ 1 & −1 & 0 \end{matrix} ]

and

B = [ \begin{matrix}0 & 1 \\ 2 & 1 \\ 1 & −1 \end{matrix} ]

Compute, if possible A^{2}, AB, B^{T} A^{T}, BA

10-11.Determine whether the matrix has an inverse. If yes, find the inverse and write the given matrix as a product of elementary matrices.

10.

A = [ \begin{matrix}1 & 1 & 1 \\ 1 & 0 & −1 \\ 1 & −1 & 1 \end{matrix} ]

11.

A = [ \begin{matrix}1 & 0 & 2 \\ 0 & 0 & −1 \\ 1 & −1 & 0 \end{matrix} ]

12. Find the inverse of

A = [ \begin{matrix}1 & 0 & 2 \\ 1 & 0 & −1 \\ 1 & −1 & 0 \end{matrix} ]

and solve Ax = b, where

b = [ \begin{matrix}1 \\ 1 \\ 5 \end{matrix} ]

13. Answer each of the following as true or false. Briefly justify your answer.

(i) A linear system of four equations in five unknowns has infinitely many solutions.

(ii)A linear system of four equations in three unknowns has no solution. (iii)If A is nonsingular then Ax = b has a unique solution. (iv) If A and B are nonsingular and of the same size then AB could be singular.

(v) If u_{1} and u_{2} are solutions to the homogeneous system Ax = 0 the u_{1} + u_{2} is also a solution to this system.

(vi) If A and B are square matrices and of the same size then AB = BA. (vii) If A and B are square nonsingular matrices and of the same size then

(AB)^{−1} = B^{−1}A^{−1}

(viii) If A and B are square nonsingular matrices and of the same size then (AB)^{T} = B^{T} A^{T}

(ix) If A is symmetric then A^{−1} is also symmetric. (x)If A^{2} is nonsingular then A^{3} can be singular. (xi) If A and B are square nonsingular matrices of the same size then A + B is

nonsingular.

14.Let A be a square matrix such that A^{3} = I. Find A^{−1}

15. For each of the following sets determine if it is a vector subspace

(a) The set of all vectors (x_{1}, x_{2}, x_{3}, x_{4}) with the property x_{1} − x_{2} + x_{3} − x_{4} = 0 (b)The set of all vectors (x_{1}, x_{2}, x_{3}, x_{4}) with the property x_{1}x_{2} − x_{3}x_{4} = 0 (c)The set of all vectors of the form v = (a + b, 2a + b, c, a + b + d) in R^{4}. (d)The set of all vectors of the form v = (a + b, 2a + b, c + 1, a + b + d) in R^{4}.

16. Explain why the following sets are not vector spaces.

a) W = {(a_{1}, a_{2}) : a_{1}, a_{2} ∈ R} (a_{1}, a_{2}) ⊕ (b_{1}, b_{2}) = (2a_{1} + 2b_{1}, 2a_{2} + 2b_{2}), c ⊙ (a_{1}, a_{2}) = (ca_{1}, ca_{2})

b) W = {(a_{1}, a_{2}) : a_{1}, a_{2} ∈ R} (a_{1}, a_{2}) ⊕ (b_{1}, b_{2}) = (a_{1} + b_{1}, 2a_{2} + b_{2}), c ⊙ (a_{1}, a_{2}) = (ca_{1}, ca_{2})

c)b) W = {(a_{1}, a_{2}) : a_{1}, a_{2} ∈ R} (a_{1}, a_{2}) ⊕ (b_{1}, b_{2}) = (a_{1} + b_{1}, a_{2} + b_{2}) c ⊙ (a_{1}, a_{2}) = (ca_{1}, 2ca_{2})

17. Express, if possible, one of the vectors of the following set as a linear combination of the others: v_{1} = (1, −1, 0), v_{2} = (0, 2, 1), v_{3}(1, −2, 3), v_{4} = (3, 6, 2).

18.For what values of d is the vector (2, 1, d) in the span of (1, 2, 1) and (2, 5, 1).

19. Let S = {v_{1}, v_{2}, v_{3}}, where v_{1} := (1, 2, 2), v_{2} := (3, 2, 1) v_{3} = (7, 6, 4). Determine if S is lineraly independent. Find a basis for the subspace W = spanS of R^{3} and its dimension . Determine whether u = (1, 6, 5) is in W.

20.For the problem (a) (b) (c) determine whether the given set of vectors is linearly inedpendent or linearly dependent:

(a) {(1, 0, −1), (2, 1, 1)}

(b) {(1, 0, −1), (2, 1, −1), (0, 1, 3)}

(c) {(1, 0, −1), (1, 3, 4), (2, 1, −1), (0, 1, 3)}

For the problem (d) (e) (f) determine whether the given set of vectors spans R^{3} (d) {(1, 1, −1), (2, 1, 1)}

(e) {(1, 0, 0), (1, 3, 4), (2, 1, 0), (0, 1, 3)}

(f) {(1, 0, 0), (1, 3, 4), (2, 1, 0)}

For the problem (g) (h) (i) determine whether the given set of vectors form a basis of R^{3}

(g)(1, 2, 0), (1, 1, 1), (1, 0, 1) . (h)(1, 2, 0), (1, 1, 1), (2, 3, 1) (i)(1, 2, 0), (1, 1, 1)

21. Let W = {(1, −1, −1, 2), (0, 1, 2, 0), (2, −1, 0, 2), (1, 0, 1, 2)} be a set of vectors in R^{4}. Find a basis of subspace of spanW consisting of some vectors in W.

22.Let W be the subspace of R^{4} consisting of all vectors of the form (b + c, a − b, 2b − a + c, a + c). Find a basis of W and its dimension.

23.Find a basis of of {(x, y, z) | x + y + 2z = 0}