Math

EC1.pdf

Math 220 Extra Credit HW Assn. 1 Name:

ID#: Section#:

I prefer that you submit your answers on a printed copy of this document, like it’s a quiz or exam. However, you may instead rewrite the questions by hand before solving them. Staple sheets together, in order. Be neat. Always give enough work and clear explanation so that fellow students could follow what you did (from start to finish) just by reading your paper. Remember that if you submit this extra credit you will not be eligible for the oral/board extra credit exams for either tests 1 or 2.

1. (a) Prove that E.R.O.s preserve solutions to linear systems of 2 equations.

(b) Discuss how you would generalize your proof to linear systems with more than 2 equations. Be sure to justify why your generalization would work.

2. Given a general solution to the homogeneous equation involving a matrix, A, in R3×5 to be

x⃗ = v⃗1x2 + v⃗2x3 + v⃗3x5

prove directly (i.e. use only definitions and no other theorems) that

(a) the second column of A is a linear combination of the first and fourth columns,

(b) the third column of A is a linear combination of the first and fourth columns, and

(d) the span of the columns of A is equal to the span of the first and fourth columns of A.

3. (a) If A ∈ R7×4, Col(A) contains 2 planes, and ∃v⃗ ∈ R7 s.t. Ax = v is inconsistent then what is the nullity of A?

(b) Could there be more than 1 right answer to (a)? Justify your responses as always.

4. (a) Define the span of a set of vectors.

(b) Define the basis for the span of a set of vectors of equal dimension.

5. Let W be a vector space in Rn with basis B of dimension q. Define f on W such that f maps the vectors in W to the vectors representing their coordinates with respect to the basis B.

(a) Prove that f defines a valid function.

(b) What is the codomain of f?

(d) Prove that f is 1-1.