How to proof:

Matrix A have a size of m×n, and the rank is r. How can we rigorous proof that the dimension of column space are always equal to the dimensional of row space?

(I can use many examples to show this work, but how to proof rigorously?)

x² y" + (x²+x) y’ +(2x-1) y = 0,

1. Find the general solution of y₁ with r₁ and calculate the coefficient up to c₄ and also find the general expression of the recursion formula, (recursion formula for y₁)
2. Find the general solution of y₂ based on theorem 4.3.1. (Hint, set d₂ = 0)

Solve the following initial-value differential equations using Laplace and inverse transformation.

y''-y=3e^(2t),   y(0)=6, y'(0)=3

For all n > 2 except n = 6, show how to arrange the numbers 1,2,...,n² in an n x n array so that each row and column sum to the same constant.

Find the first four terms of the sequence (an)n≥1 with the given definition. Determine if they are potentially arithmetic or geometric.

(a) an is the number of n-bit strings which have more 1’s than 0’s. (Also, write down the strings for n ≤ 4.)

(b) an is the number of n-bit strings in which the number of 1’s is greater than or equal to the number of 0’s in every prefix. For example, 010111 would not qualify, since the prefix 010 has more 0’s than 1’s. (Also, write down the strings for n ≤ 4.)

(c) an is the number of lattice paths from (0, 0) to (n, n). (Refer to Section 1.2 if you need a refresher on lattice paths.)

et A be a 157 x 157 upper-triangular matrix. Suppose that every diagonal entry of A is 1 and that there is at least one nonzero off-diagonal entry in A. Is A diagonalizable? Explain how you can answer this question mentally, with no non-trivial calculations.

Let L₁ be the line passing through the point P₁=(−3, −1, 1) with direction vector →d₁=[1, −2, −1]^T, and let L₂ be the line passing through the point P₂=(8, −3, 1) with direction vector →d₂=[−1, 0, 2]^T.
Find the shortest distance d between these two lines, and find a point Q₁ on L₁ and a point Q₂ on L₂ so that d(Q₁,Q₂) = d. Use the square root symbol '√' where needed to give an exact value for your answer.

Q₁ =
Q₂ =

Let A be a set with m elements and B a set of n elements, where m; n are positive integers. Find the number of one-to-one functions from A to B.

Convert to standard maximum form and apply two iterations of simplex process using slack form.

Maximize

2x1 -6x3

Subject to

x1 + x2 – x3 <= 7

3x1 – x2 >= 8

-x1 + 2x2 + 2x3 >= -2

x2, x3 >=0

Please write the answer very clearly.

Find general solutions of the following systems using undetermined coefficients.

X′ =(2x2 matrix) ( 2 2; 3 1 ) X + (column matrix) ( e^−4t; 0 )

1. Find general solutions of the following systems using undetermined coefficients.

a.) x′ = −5x + 6y + 1

y′ = −7y + t .

b.) x′ = 6x − 5y + e^5t

y′ = x + 4y .

c.) x′ = −6x − 3y + te^2t

y′ = 4x + y .