代做Math 350、代写CS/python设计、代做null-space留学生

日期：2018-10-25 10:03

Problem Set 5 parts I,II, Math 350, Fall 2018, Section 4. Due Tuesday Oct 23

(1) Find all solutions f ∈ P3 to f(1) = 1 and f(2) = 2, using Gaussian elimination,

in the following steps.

(a) Write the system of equations f(1) = 1 and f(2) = 2 as an augmented matrix

[A|b].

(b) Find a row echelon form and reduced row echelon form of [A|b].

(c) Find a basis for the null-space of A.

(d) Find a basis for the column-space of A.

(e) Find a basis for the row-space of A.

(2) (a) Given an example of elementary matrices E1, E2 so that E1E2 is also elementary.

(b) Give an example of elementary matrices E1, E2 so that E1E2 is not elementary.

(3) (a) Given an example of two different row echelon forms for the matrix A from

Problem (1).

(b) Explain why the reduced row echelon form for the matrix A in Problem (1) is

unique. (Hint: Suppose that R1 and R2 are matrices in row-echelon form equivalent

to A. Show that R1 has to equal R2, using the relation between the row-spaces or

null-spaces of R1 and R2.)

(4) Let A be a square n × n matrix.

(a) True or false: The dimension of the column space of A2

is the same as the

dimension of the column space of A. Explain your answer.

(b) True or false: The dimension of the null-space plus the dimension of the rowspace

is equal to n. Explain your answer.

(5) Let v be a non-zero column vector. Show that the matrix A = vvT has rank

one.

(6) Use Gaussian elimination to solve the system of linear equations x1 + 2x2 ?x3 +

3x4 = 2 2x1 + 4x2 x3 + 6x4 = 5 x2 + 2x4 = 3. Show your work and for each step

indicate the elementary row operation.

(7) (a) Use Gaussian elimination to find the inverse of the matrix

1 2 1

1 1 2

1 0 1.

For each step indicate which elementary row operation you used. (b) Write the

matrix

0 1 0

0 0 1

1 0 0

as the product of elementary matrices.