Standard Form Of A Linear Equation Worksheet - Math 205b Quiz 03, 2010
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1
2Quiz 03 page 1
01/29/2010
Math 205B
Name
2
1. Let u, v, z, b and k be as in the following figure. All these vectors belong to R
and all are drawn in “standard
position”.
1A. Find all ways to express b as a linear combination of u and v.
1B. Find all solutions of x
u + x
v = k.
1
2
1C. Find one way to write z as a LC of u and v.
1D. Use your answer to 1C to find all solutions of x
u + x
v + x
z = b in parametric vector form x = p + v
where p is a
1
2
3
h
particular solution of x
u + x
v + x
z = b and v
represents all solutions of the homogeneous equation x
u + x
v + x
z = 0.
1
2
3
h
1
2
3
(Take x
as free). (Hint: From 1A you know a particular solution αu + βv = b. You can add x
(z
z) to the left side of
3
3
this because it would be “adding nothing”. Now rewrite the second z as in 1C and simplify).
NOTE: This quiz continues on the other side.
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