Study Sheet — Exam 1Math 45 — Linear AlgebraDavid ArnoldFall 1997InstructionsYou may not use calculators nor Matlab on this exam. Place your solution to each problem onyour own paper.1. Let u =u1u2be an arbitrary vector from R2. Let c and d be arbitrary real numbers.Prove: Ýc + dÞu = cu + du.2. Consider the following system of linear equations:x1? x2? 2x3= 02x1? x3= ?2?x1? x2+ 4x3= ?8a. Set up an augmented matrix representing this system of equations.b. Use the elementary row operations to place your augmented matrix in row echelon form.c. Use back substitution and the row echelon form of the augmented matrix to find the solutionof the system of equations.3. Each of the following matrices represent the reduced row echelon form the augmented for thefollowing system of linear equations with unknowns x1, x2, u, xn.a11x1+ a12x2+ ` + a1nxn= b1a21x1+ a22x2+ ` + a2nxn= b2_am1x1+ am2x2+ ` + amnxn= bmState the solution to each system. That is, clearly state the value of each of the unknown variablesx1, x2, u, xn.a.1 0 0 0 ?20 1 0 0 ?40 0 1 0 120 0 0 0 3b.1 0 1 0 ?20 1 ?1 0 30 0 0 1 ?50 0 0 0 01c.1 0 ?20 1 10 0 00 0 0d.1 ?1 0 ?1 0 5 00 0 1 2 0 1 10 0 0 0 1 ?1 ?24. Find the equation of a second degree polynomial passing through the points Ý?1,1Þ,Ý1,2Þ, andÝ2,4Þ.5. Jaime has 12 coins in her pocket. All of the coins are nickels, dimes, or quarters. The total valueof the coins is $1.65.a. Let n, d, and q represent the number of nickels, dimes, and quarters in Jaime’s pocket,respectively. Set up a system linear equations modeling the problem statement.b. Set up the augmented matrix for your system of equations. Use the elementary rowoperations to place the augmented matrix in reduced row echelon form. Clearly state thesolution of the system of equations.c. List (in an organized manner) all possible combinations of coins that satisfy the problemstatement.6. Consider the vectors u and v in Figure 1.Draw a sketch indicating how to construct the linear combination 2u ? 3v.7. Define Rn.8. Define what is meant by a linear combination of the vectors v1, v2,u, vn.9. Define the span of the vectors v1, v2,u, vn.10. Define what is meant when one says that the vectors v1, v2,u, vnare linearly dependent.11. Define what is meant when one says that the vectors v1, v2,u, vnare linearly independent.12. Consider the following system of linear equations:2x1+ 2x2+ 3x3= 4?x1+ x3= 0x1+ x2= ?2?2x1? 2x2+ x3= 4a. Set up a vector equation for this system of equations.b. Set up a matrix equation for this system of equations.13. Find each of the following matrix-vector products, if possible.a.1 2?1 10 21?12b.1 ?1 00 1 20 0 ?33 2 ?21?1214. Consider the following vectors:a1=1?21, a2=?101, u =h2?1For what values of h is the vector u in the span of the vectors a1and a2?15. Show that the vectorsv1=1?21, v2=011, v3=2?7?1are dependent by finding at least one particular non-trivial linear combination of v1, v2, and v3equaling the zero vector. Clearly show that your solution checks, showing that the vectors aretherefore dependent.16. Describe geometrically the span of each of the following sets of
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