MT210 Midterm 1 Sample Exam: Linear Algebra, Exams of Linear Algebra

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MT210 MIDTERM 1 SAMPLE 4

İLKER S. YÜCE

FEBRUARY 16, 2011

Surname, Name:

QUESTION 1. SYSTEMS OF LINEAR EQUATIONS

Solve the linear system x 1 + x 2 + x 3 = 4 −x 1 − x 2 + x 3 = − 2 2 x 1 − x 2 + 2 x 3 = 2

QUESTION 2. ROW REDUCTION AND ECHELON FORMS

Find the row reduced echelon form of the matrix below and mark the pivot positions:       1 − 2 − 4 3 2 5 − 2 9 1 7 2 6 0 5 − 2 9 1 − 2 − 4 3

QUESTION 4. MATRIX EQUATIONS

Consider the linear system

3 x 1 + 2 x 2 − x 3 − x 4 = − 3 −x 1 + x 3 + 2 x 4 = 1 2 x 1 + 2 x 2 + x 4 = − 2 x 1 + 2 x 2 + x 3 + 3 x 4 = − 1

A. Write the linear system in the matrix form A x = b.

B. Solve the matrix equation A x = b and write the solution in parametric-vector form.

QUESTION 5. LINEAR INDEPENDENCE

Let v 1 =

, v 2 =

, v 3 =

 (^) and v 4 =

A. Show that S = { v 1 , v 2 , v 3 , v 4 } is linearly dependent. B. Show that T = { v 1 , v 2 , v 3 } is linearly independent. C. Show that v 4 can be written as a linear combination of v 1 , v 2 , v 3.

QUESTION 7. TRUE OF FALSE

Mark each statement True or False. Justify your answer. Let S be a set of m vectors in R n.

A.) If m > n then S is linearly independent.

B.) If the zero vector is in S , then S is linearly dependent.

C.) If S is linearly independent and T is a subset of S , then T is linearly independent.

D.) If T is linearly dependent and T is a subset of S , then S is linearly dependent.

E.) The linear system A x = b has a unique solution if and only if the column vectors of A are linearly independent.