Linear Algebra: I need detailed worked out solutions to the following attached problems, so I can compare whether I'm doing things right (my prof will provide later on, after I turn it in, but that does me no good after the fact). Basically provide a solution guide. If I like your work, I'll likely hire you again for future solutions in this material, in the next coming 10 weeks (10 weekly assignments). Thanks.

 

 

Homework 1
due Sept 29
1. Describe geometrically (line, plane, three-dimensional space) the span of 1
2
1
1
1
0
1 1
0
a) 2 and 4
b) 1 and −1
c) −1 and
and
3
6
0
0
0
−1
−1 1
0
1
d) −1 and 1 and 0
0
−1
1
Recall: The span of vectors v and w is the set of all possible linear combinations c v +d w
as c and d range over all real numbers. More generally, the span of a set of vectors v1 , v2 ..., vn
is the set of all possible linear combinations c1 v1 + c2 v2 + · · · cn vn .
 
 
 
 
3
1
3
1
2. a) If v + w =
and w =
, what is v? b) If v + w =
and v − w =
,
2
1
2
1
what are v and w?
3. Consider the vectors 1 u = 2
3 1
v = −2
1 1 w = 4 .
4 Find c, d, e such that c u + d v + e w = 0. Describe geometrically the span of these 3 vectors.
 
 
−3
4
4. Suppose u =
and v =
. What
2
−3  
1
there exist c and d such that c u + d v =
?
0
 
 
−3
6
5. Suppose u =
and v =
. What
2
−4  
1
there exist c and d such that c u + d v =
?
0 (geometrically) is the span of u and v? Do
If so, find c and d. (geometrically) is the span of u and v? Do
If so, find c and d. From Strang Problem Set 1.1: problems 13, 14.
6. Consider the three vectors in part (c) of problem 1. Find a single vector v such that
its inner product with every one of the vectors in 1(c) vanishes. What is the dot product
of this v with an arbitrary linear combination of the vectors in 1(c)? What is the sum of
components of an arbitrary linear combination of the vectors in 1(c)?
1 7. Consider the three vectors from problem 3. Find u · v, u · (v − w), v · w, u · w. Check
that u · (v − w) = u · v − u · w.
Recall: The dot product is a bilinear map from two copies of Rn to R. This means that
it takes two vectors and spits out a real number, while respecting the additive and scalarmultiplicative properties of both vectors simultaneously. Algebraically, this means that for
any u, v, w we have
(u+v)·w = u·w+v·w and u·(v+w) = u·v+u·w and (c u)·v = c (u·v) = u·(c v).
In addition, the dot product is symmetric. This means that for any u, v
u · v = v · u. 8. Take the two vectors u and v from problem 4. Check that (2 u − v) · v = 2(u · v) − v · v
and that (u − v) · (u + v) = ||u||2 − ||v||2 .
9. What is (c u + d v) · (e w), expanded out? What is (c u + d v) · (e w + f t), expanded out? 2

Attachments:

• 1492566081-hw1.pdf