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MAT 202, Homework #4, Summer 2016 Name _______________________________________________
Instructions: Write your work up neatly and attach to this page. Use exact values unless specifically
asked to round. Show all work.
1
−3
1. Sketch the vectors 𝑢
⃗ = [ ],𝑣 = [ ],𝑢
⃗ + 𝑣 and explain how this illustrates the parallelogram
4
2
rule.
1
−3
0
2. Sketch the vectors 𝑢
⃗ = [4] , 𝑣 = [ 2 ] , 𝑤
⃗⃗ = [−1] , 𝑢
⃗ +𝑣+𝑤
⃗⃗ .
2
−3
5
3. For the vectors 𝑎 = [
2
1
4
−1 ⃗
3
] , 𝑏 = [ ] , 𝑐 = [7] , 𝑑 = [−1] , 𝑒 = [−5], find the following:
2
5
1
0
2
a. 𝑎 − 𝑏⃗
b. 3𝑏⃗ + 2𝑎
c. 𝑐 + 2𝑑 − 4𝑒
4. For each statement below determine if it is true or false. If the statement is false, briefly explain
why it is false and give the true statement.
a. If f is a function in the vector space V of all real-valued functions on R and if f (t )  0 for
some t, then f is the zero vector in V.
b. A vector is an arrow in three-dimensional space.
c. A subset H of a vector space V is a subspace of V if the zero vector is in H.
d. A subspace is also a vector space.
e. A vector is any element of a vector space.
f.
𝑅 2 is a subspace of 𝑅 3.
g. A subset H of a vector space V is a subspace of V if the following conditions are satisfied: i)
the zero vector of V is in H, ii) u , v and u  v are in H, and iii) c is a scalar and cu is in H.
 2 
 5 
and   lie on a line through the origin.

5
2
h. The points in the plane corresponding to 
i.
An example of a linear combination of vectors v1 and v2 is the vector
j.
Any list of 5 real numbers is a vector in 𝑅 5.
1
v1 .
2
1
1
0
2
0
5
2
2
1
1
2
3
5. Write 𝑣 = −11 as a linear combination of −3 , 0 , 1 , −1 , 2 . Is the solution
4
2
1
2
−1
11
[−1] [1] [−4] [ 1 ] [−1]
[ 9 ]
unique?
6. For each of the sets below, determine if the set is a vector space or subspace.
𝑎
a. 𝐻 = {[ 2 ] , 𝑎, 𝑏 𝑟𝑒𝑎𝑙}
𝑏
𝑎
b. 𝑉 = {[𝑏 ] , 𝑎 = 𝑏 + 𝑐}
𝑐
𝑎 2
c. 𝑇 = {[
] , 𝑎, 𝑏 𝑟𝑒𝑎𝑙}
0 𝑏
d. C = {𝑡ℎ𝑒 𝑠𝑒𝑡 𝑜𝑓 𝑎𝑙𝑙 𝑐𝑜𝑚𝑝𝑙𝑒𝑥 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑓𝑜𝑟𝑚 𝑎 + 𝑏𝑖, 𝑤ℎ𝑒𝑟𝑒 𝑎, 𝑏 𝑎𝑟𝑒 𝑟𝑒𝑎𝑙}
e. 𝐽 = {𝑡ℎ𝑒 𝑠𝑒𝑡 𝑜𝑓 𝑎𝑙𝑙 𝑝𝑜𝑙𝑦𝑛𝑜𝑚𝑖𝑎𝑙𝑠 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝑝(𝑡)𝑑𝑖𝑣𝑖𝑑𝑒𝑠 𝑒𝑣𝑒𝑛𝑙𝑦 𝑏𝑦 (𝑡 − 1)} [Hint: write
p(t) in factored form, with the factor (t-1) pulled out. What does the other factor look like?]
f. 0 = {𝑡ℎ𝑒 𝑠𝑒𝑡 𝑜𝑓 𝑎𝑙𝑙 𝑜𝑑𝑑 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛𝑠: 𝑓(−𝑥) = −𝑓(𝑥)}
g. 𝑊 is the set of all nxn matrices such that 𝐴2 = 𝐴.
h. 𝑄 is the set of all exponential functions
i. 𝑆 is the set of all nxn singular matrices
7. For each of the alternate definitions of addition or scalar multiplication on 𝑅, 𝑅 2 , 𝑜𝑟 𝑅 3,
determine whether the resulting set is a vector space. In some cases, only one of the definitions
for vector addition or scalar multiplication is altered. In those cases, you need only check the
affected properties. In cases where both are altered, you should check all properties. Be sure to
state clearly any properties that fail.
𝑥1
𝑥2
𝑥1
𝑥 +𝑥
√𝑐𝑥1
a. [𝑦 ] + [𝑦 ] = [𝑦1 + 𝑦2 ] , 𝑐 [𝑦 ] = [
] 𝑖𝑛 𝑅 2
1
2
1
1
2
√𝑐𝑦1
𝑥1
𝑥2
𝑥1
𝑥1 + 𝑥2 + 1
𝑐𝑥1 + 𝑐 − 1
𝑦
𝑦
𝑦
𝑦
+
𝑦
+
1
𝑐𝑦
b. [ 1 ] + [ 2 ] = [ 1
] , 𝑐 [ 1 ] = [ 1 + 𝑐 − 1] 𝑖𝑛 𝑅 3
2
𝑧1
𝑧2
𝑧1
𝑧1 + 𝑧2 + 1
𝑐𝑧1 + 𝑐 − 1
𝑥1
𝑥2
𝑥1 𝑥2
𝑥1
𝑐𝑥1
c. [𝑦 ] + [𝑦 ] = [𝑦 𝑦 ] , 𝑐 [𝑦 ] = [𝑐𝑦 ] 𝑖𝑛 𝑅 2
1
2
1 2
1
1
d. 𝑥 + 𝑦 = 𝑥𝑦, 𝑐𝑥 = 𝑥 𝑐 𝑖𝑛 𝑅 + (positive real values only).
The ten properties of a vector space are:
i.
𝑢
⃗ + 𝑣 in V for every 𝑢
⃗ ,𝑣 𝜖 𝑉
ii.
𝑢
⃗ +𝑣 =𝑣+𝑢
⃗
iii.
𝑢
⃗ + (𝑣 + 𝑤
⃗⃗ ) = (𝑢
⃗ + 𝑣) + 𝑤
⃗⃗
⃗
iv.
V has a vector 0 such that for every 𝑢
⃗ in V, 𝑢
⃗ + ⃗0 = 𝑢
⃗
v.
For every 𝑢
⃗ in V, there is a vector in V denoted by −𝑢
⃗ such that 𝑢
⃗ + (−𝑢
⃗ ) = ⃗0
vi.
𝑐𝑢
⃗ in V for every 𝑢
⃗ in V, and every 𝑐 𝜖 𝑅
vii.
𝑐(𝑢
⃗ + 𝑣 ) = 𝑐𝑢
⃗ + 𝑐𝑣
(𝑐 + 𝑑)𝑢
viii.
⃗ = 𝑐𝑢
⃗ + 𝑑𝑢
⃗
(𝑐𝑑)𝑢
ix.
⃗ = 𝑐(𝑑𝑢
⃗)
x.
1𝑢
⃗ =𝑢
⃗