MATH 315 (081) ASSIGNMENT 2
Please hand in Monday February 11 at the beginning of class.
1. Here is another vector space proof. Use the vector space properties and properties of real numbers
to justify every step. Explicitly state which vector space property is being used at each step.
Let X,Y,Zand Wbe elements belonging to some vector space V. Suppose that these vectors
satisfy the relations 2X=Y+Zand W=X+Y. Prove that {Y, Z, W }is a dependent subset of
V.
2. Let [a b]tbe any vector in R2. Solve an appropriate system of linear equations to show that
[a b]tbelongs to the span S, where
S=Ω∑ 1
−1∏,∑2
2∏æ
3. Do Section 3 Exercises 3(d) and 4(d) in detail and show your work.
(a) Write down the system of equations that has the given matrix as its augmented matrix. Also,
write down the coefficient matrix for this system.
(b) Find the reduced echelon form (RREF) for the augmented matrix. Show all of your row opera-
tions.
(c) Use the RREF to identify the pivot columns. Which variables in your system are the pivot
variables?
(d) Write down the solution set of the system in vector form, showing the translation vector and
the spanning vector(s).
4. Write down all possible reduced echelon forms for a 3 ×3 matrix.
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