Linear Algebra Answers

The sum of the digits of a three-digit number is 11. If the first and last digits are interchanged, the new number is greater than the original number by 594. If the tens and the units digits are interchanged, the resulting number is greater than the original by 36. Find the original number.

A man wishes to fence in a rectangular lot. If he uses material costing P17/meter for the front of the lot and material costing P14/meter for the other three sides, the fence will cost P8,900. If he uses the cheaper material for all four sides, the fence will cost P8,540. Find the dimensions of the lot.

A man rows 12 miles downstream on a river in 2 hrs. He requires another 3 hrs. to row back to his starting position. Find the speed of the current and the rate the man can row in still water.

Two pipes empty a vat in 1hr and 12 min. When one pipe is used to fill and one to empty the vat, it is filled in 3 hrs. How long would it take each pipe to empty the vat if only one pipe is open?

Determine whether W = {(x, y,z) | x + y + z + 1 = 0, x, y,z ∈ R} a subspace of R³ or

not?

Which of the following linear systems is homogeneous. I) 3x + 14y = 0 & 2x − 3y = 7 II) 7x + 2y = 10 & 11x − 0y = 7 III) 4x − 23y = 0 & 7x − 3y = 0

Prove that if U and V are subspaces of Rn

so is U+V

2x + y = 7

x - 2y = 1

A. Write the equation in matrix form.

B. Determine the inverse of the matrix

C. Hence solve the equations.

D. x and y are matrices

"X=\\begin{bmatrix}\n 1 & 5 \\\\\n 3 & 7\n\\end{bmatrix} \n \n\n Y=\\begin{bmatrix}\n 3 & 4 \\\\\n 2 & 1\n\\end{bmatrix}"

Evaluate X² + Y

        [ 1 0 -1

3. Consider the matrix A =  0 3 0

                      -1 0 1 ]

1. Find the eigenvalues of A.
2. Find the eigenspaces corresponding to each eigenvalue from A.

2. Consider a linear transformation T: R³ → R³ defined by

  ([x         [ x + 4y +3z  

T  y     =      -5y - 4z 

    z])        5x + 10y + 7z ]

Note: T is a 3x1 matrix containing x, y, z respectively. T is equal to another matrix as shown above.

a) Find the matrix A for T

b) Find a basis for ker(T) and the dim(ker(T)). Then find dim(Im(T)), without finding a basis for Im(T). (Show all working)

c) Find a basis for Im(T)