Linear Algebra (Math 2890) Practice Problems 1

Topics for Midterm I:1.1-1.5, 1.7, 1.8

1. Show that A = [ \begin{matrix}1 & 2 & −2 & 0 & 7 \\ −2 & −3 & 1 & −1 & −5 \\ −3 & −4 & 0 & −2 & −3 \\ 3 & 6 & −6 & 5 & 1 \end{matrix} ] is row equivalent to [ \begin{matrix}1 & 0 & 4 & 0 & −3 \\ 0 & 1 & −3 & 0 & 5 \\ 0 & 0 & 0 & 1 & −4 \\ 0 & 0 & 0 & 0 & 0 \end{matrix} ].

2. Determine if the following systems are consistent and if so give all solutions in parametric vector form. (a)

x_{1} −2x_{2} = 3
2x_{1} −7x_{2} = 0
−5x_{1} +8x_{2} = 5

(b)

\begin{aligned}x_{1} & +2x_{2} & −3x_{3} & +x_{4} & = 1 \\ −x_{1} & −2x_{2} & +4x_{3} & −x_{4} & = 6 \\ −2x_{1} & −4x_{2} & +7x_{3} & −x_{4} & = 1 \end{aligned}

3. Let A = [\begin{matrix}1 & 3 & −4 & 7 \\ 2 & 6 & 5 & 1 \\ 3 & 9 & 4 & 5 \end{matrix}].

(a) Find all the solutions of the non-homogeneous system Ax = b,

and write them in parametric form, where b = [\begin{matrix}−1 \\ −2 \\ −3 \end{matrix}].

(b) Find all the solutions of the homogeneous system Ax = 0, and write them in parametric form.

(c) Are the columns of the matrix A linearly independent? Write down a linear relation between the columns of A if they are dependent.

4. Let S = Span\{ [ \begin{matrix}1 \\ −2 \\ 3 \\ 1 \end{matrix} ] , [ \begin{matrix}0 \\ 1 \\ 1 \\ −2 \end{matrix} ] , [ \begin{matrix}1 \\ −3 \\ 2 \\ 3 \end{matrix} ] , [ \begin{matrix}0 \\ 1 \\ 1 \\ −4 \end{matrix} ] \}.

(a) Find all the vectors u = [\begin{matrix}a \\ b \\ c \\ d \end{matrix}] such that the u is in S. Write these

u in parametric form. Justify your answer.

(b) Is v = [ \begin{matrix}−1 \\ 3 \\ −2 \\ 1 \end{matrix} ] in S.

(c) Is w = [ \begin{matrix}1 \\ 3 \\ −2 \\ 1 \end{matrix} ] in S.

5. Consider a linear system whose augmented matrix is of the form

[ \begin{matrix}1 & 1 & 1 & 2 \\ 1 & 2 & 1 & b \\ 3 & 5 & a & 1 \end{matrix} ]

(a) For what values of a will the system have a unique solution? What is the solution?(your answer may involve a and b)

(b) For what values of a and b will the system have infinitely many solutions?

(c) For what values of a and b will the system be inconsistent?

6. Determine if the columns of the matrix form a linearly independent set. Justify your answer.

[\begin{matrix}−2 & 1 \\ 4 & −2 \\ 0 & 0 \\ −6 & 3 \end{matrix}], [\begin{matrix}−2 & 1 \\ 4 & −2 \\ 2 & 2 \end{matrix}], [\begin{matrix}1 & 1 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 0 & 0 \end{matrix}] , [\begin{matrix}1 & 1 & 3 \\ 0 & 1 & 2 \\ 1 & −1 & −1 \\ 1 & 0 & 1 \end{matrix}], [\begin{matrix}−4 & −3 & 1 & 5 & 1 \\ 2 & −1 & 4 & −1 & 2 \\ 1 & 2 & 3 & 6 & −3 \\ 5 & 4 & 6 & −3 & 2 \end{matrix}].

7. M = [\begin{matrix}1 & 1 & 2 \\ 1 & a + 1 & 3 \\ 1 & a & a + 1 \end{matrix}].

(a) Describe the values of a so that the column vectors of M are linearly independent.

(b) Describe the values of a so that the column vectors of M are linearly dependent.

8. Let e_{1} = [\begin{matrix}1 \\ 0 \\ 0 \end{matrix}], e_{2} = [\begin{matrix}0 \\ 1 \\ 0 \end{matrix}] and e_{3} = [\begin{matrix}0 \\ 0 \\ 1 \end{matrix}]. Suppose T : R^{3} ↦ R^{2} is a

linear transformation such that T (e_{1} + e_{2}) = [\begin{matrix}1 \\ −1 \end{matrix}] , T (e_{1} − e_{2}) = [\begin{matrix}2 \\ 3 \end{matrix}]

and T (e_{1} + e_{2} + e_{3}) = [\begin{matrix}1 \\ −2 \end{matrix}]. What is T ([\begin{matrix}1 \\ 2 \\ 3 \end{matrix}])?