Linear Algebra:Â 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 againÂ for future solutions in this material, in the next coming 10 weeks (10 weekly assignments). Thanks.

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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 âˆ’1

c) âˆ’1 and

and

3

6

0

0

0

âˆ’1

âˆ’1 1

0

1

d) âˆ’1 and 1 and 0

0

âˆ’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 + Â· Â· Â· cn vn .

3

1

3

1

2. a) If v + w =

and w =

, what is v? b) If v + w =

and v âˆ’ w =

,

2

1

2

1

what are v and w?

3. Consider the vectors 1 u = 2

3 1

v = âˆ’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.

âˆ’3

4

4. Suppose u =

and v =

. What

2

âˆ’3

1

there exist c and d such that c u + d v =

?

0

âˆ’3

6

5. Suppose u =

and v =

. What

2

âˆ’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 Â· v, u Â· (v âˆ’ w), v Â· w, u Â· w. Check

that u Â· (v âˆ’ w) = u Â· v âˆ’ u Â· 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)Â·w = uÂ·w+vÂ·w and uÂ·(v+w) = uÂ·v+uÂ·w and (c u)Â·v = c (uÂ·v) = uÂ·(c v).

In addition, the dot product is symmetric. This means that for any u, v

u Â· v = v Â· u. 8. Take the two vectors u and v from problem 4. Check that (2 u âˆ’ v) Â· v = 2(u Â· v) âˆ’ v Â· v

and that (u âˆ’ v) Â· (u + v) = ||u||2 âˆ’ ||v||2 .

9. What is (c u + d v) Â· (e w), expanded out? What is (c u + d v) Â· (e w + f t), expanded out? 2

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# Answers

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**Posted 19 Apr 2017 01:04 AM**
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