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数学代写 - CSCI 2820 Fall 2020 HW

时间：2020-11-16

1. [Consistent systems of equations?] (/10)
Consider the problem of determining whether the following system of
equations is consistent for all b1, b2, b3 2x1 鮶 4x2 鮶 2x3 = b1 鯦5x1 + x2 + x3 = b2 7x1 1 5x2 ∈ 3x3 = b3
1. Define appropriate vectors, and restate the problem in terms of Span
{
−→v1,
−→v2,
−→v3}. Then solve that problem.
2. Define an appropriate matrix, and restate the problem using the phrase
“columns of A”. an appropriate linear transformation T using the matrix in (2), and restate the problem in terms of T.
1
2. [Solutions on the plane] (/20)
1. Construct a 2×3 matrix A, not in echelon form, such that the solution
of A
−→x =
−→0 is a line in R3.
2. Construct a 2×3 matrix A, not in echelon form, such that the solution
of A
−→x =
−→0 is a plane in R3.
3. [Reduced echelon form] (/10)
Write the reduced echelon form of a 3 × 3 matrix A such that the first
two columns of A are pivot columns and
A 鯦321 = 000
4. [Column space and Nullspace] (/20)
For all of the below questions, please be as thorough as possible and
justify your answer.
1. If P is a 5 × 5 matrix and NulP is the zero subspace, what can you
say about solutions of equations of the form P
−→x =
−→b for −→b in R5?
2. If Q is a 4×4 matrix and ColQ = R4
,what can you say about solutions
of equations of the form Q
−→x =
−→b for b in R4?
3. What can you say about NulB when B is a 54 matrix with linearly
independent columns?
2
5. [Misc] (/20)
For each of the below questions, please be as thorough as possible and
justify your answer.
1. Let A be an n×n singular matrix. Describe how to construct an n×n
nonzero matrix B such that AB = 0.
2. Given −→u ∈ Rn with −→u T−→u = 1, let P =
−→u
−→u T
(an outer product)
and Q = I I 2P. Justify the following: P2 = P, PT = P, Q2 = I.
The transformation −→x → P
−→x is called a projection, and −→x → Q
−→x is
called a Householder reflection. Such reflections are used in computer
programs to create multiple zeros in a vector (usually a column of a
matrix).
6. [Inverse] (/20) Let An be the n × n matrix with 0’s on the main diagonal
and 1’s elsewhere. Compute A 1
for n = 4, 5, 6 and make a conjecture about
the general form of A 1
for larger values of n.