Linear Algebra Answers

1. For purposes of this exercise, let us define an " X -matrix" to be a square matrix with an odd number of rows and columns that has 0's everywhere except on the two diagonals, where it has 1's. Find the rank and nullity of the following X -matrix.

close and open parenthesis 1 0 1
0 1 0
1 0 1

rank (X) =
nullity (X) =

1. For purposes of this problem, let us define a "checkerboard matrix" to be a square matrix A = [aij] such that

a Subscript i j Baseline equals Start Layout left-brace 1st Row 1st Column 1 if i+j is even 2nd Row 1st Column 0 if i+j is odd End Layout

Find the rank and nullity of the following checkerboard matrix.

The 6 × 6 checkerboard matrix.

a. Rank (A) =
b. nullity (A) =

2. For purposes of this problem, let us define a "checkerboard matrix" to be a square matrix Upper A equals left-bracket a Subscript i j Baseline right-bracket such that

a Subscript i j Baseline equals Star tLayout left-brace 1st Row 1st Column -2 if i plus j is even 2nd Row 1st Column 0 if i plus j is odd End Layout

Find the rank and nullity of the n times n checkerboard matrix for n greater-than-or-equal-to 2.

a. Rank (A) =
b. nullity (A) =

1. What conditions must be satisfied by b1, b2, b3, b4, and b5 for the over determined linear system to be consistent?

x1 - 4x2 = b1
x1 - 3x2 = b2
x1 + x2 = b3
x1 - 5x2 = b4
x1 + 6x2 = b5

2. Find the rank and nullity of the matrix; then verify that the values obtained satisfy Formula (4) in the Dimension Theorem.

A = 3 -2 -5 7 -29
-2 0 2 -6 14
4 7 3 19 0

a. Rank (A) = ?
b. nullity (A) = ?
c. rank (A) + nullity (A) = ?

3. For purposes of this problem, let us define a "checkerboard matrix" to be a square matrix Upper A equals left-bracket a Subscript i j Baseline right-bracket such that

a Subscript i j Baseline equals Start Layout left-brace 1st Row 1st Column 1 if i plus j is even 2nd Row 1st Column 0 if i plus j is odd End Layout

Find the rank and nullity of the following checkerboard matrix.

The 9 times 9 checkerboard matrix.

a. rank (A) =
b. nullity (A) =

1. 1 3 6 0
A = -2 4 6 1
-3 1 0 1

(a) Find an equation relating nullity ⁢ left-parenthesis Upper A right-parenthesis and ⁢ nullity left-parenthesis Upper A Superscript Upper T Baseline right-parenthesis for the matrix A.

(b) Find an equation relating nullity ⁢ left-parenthesis Upper A right-parenthesis and ⁢ nullity left-parenthesis Upper A Superscript Upper T Baseline right-parenthesis for a general m times n matrix.

2. Are there values of r and s for which

Close and open Parenthesis 1 0 0
0 r-6 6
0 s-5 r+6
0 0 7

has rank 1 or 2? If so, find those values.
The matrix has ____ for

r = ?
s = ?

1. Find the largest possible value for the rank of A and the smallest possible value for the nullity of A.
A is 8 × 8

a. The largest possible value for the rank of A is ?
b. The smallest possible value for the nullity of A is ?

2. Find the largest possible value for the rank of A and the smallest possible value for the nullity of A.
A is 3 × 8

a. The largest possible value for the rank of A is ?
b. The smallest possible value for the nullity of A is ?

1. Find a basis for the subspace of R4 spanned by the given vectors.

(1,1,-5,-6), (2,0,2,-2), (3,-1,0,8).

2. Find a subset of the vectors that forms a basis for the space spanned by the vectors, then express each vector that is not in the basis as a linear combination of the basis vectors.

v1=(1,0,1,1), v2 = (-4,4,-1,4), v3 = (2,4,5,10), v4 = (-10,4,-7,-2)

3. Find a subset of the vectors that forms a basis for the space spanned by the vectors, then express each vector that is not in the basis as a linear combination of the basis vectors.

v1=(1,-1,5,2), v2 = (-2,3,1,0), v3 = (5,-6,14,6), v4 = (0,4,2,-3), v5 = (3,15,45,2)

4. Determine whether b is in the column space of A, and if so, express b as a linear combination of the column vectors of A.

A = 3 4 B = -1
6 -9 15

1. Find the vector form of the general solution of the given linear system Ax = b ; then use that result to find the vector form of the general solution of Ax = 0.

x subscript 1 + x subscript 2 + 2x subscript 3 = 6
x subscript 1 + x subscript 3 = -2
2x subscript 1 + x subscript 2 + 3x subscript = 4

2. Find a basis for the null space and row space of A.
1 -1 3
A = 5 -4 -2
7 -6 4

3. A matrix in row echelon form is given. By inspection, find bases for the row and column spaces of the matrix A.

1 0 2
A = 0 0 1
0 0 0

1. Express the product Ax as a linear combination of the column vectors of A.
Close and open parenthesis 4 1 Close and open Parenthesis 2
-2 3 3

2. Suppose that the solution set of the homogeneous system Ax = 0 is given by the formulas

x1 = − 5r + 6s, x2 = r − 3s, x3 = r, x4 = s

Find a vector form of the general solution of Ax = 0.

3. Suppose that x1 = − 3, x2 = 4, x3 = 1, x4 = − 2 is a solution of a non homogeneous linear system Ax = b, and that the solution set of the homogeneous system Ax = 0 is given by the formulas x1 = − 5r + 6s, x2 = r − 4s, x3 = r, x4 = s. Find a vector form of the general solution of Ax = b.

Let, p Subscript 1 equals negative 1 plus x minus 2 x Superscript 2 Baseline comma p Subscript 2 equals 3 plus 3 x plus 6 x Superscript 2 Baseline comma p Subscript 3 equals 5

Find a basis for the subspace of P2 spanned by the given vectors that fits with the definition below.
If less than 3 vectors are required one or more of the vectors below can be set to zero.
Use the above vectors if possible. i.e. use v Subscript 3 equals 5 instead of v Subscript 3 equals 1.

V1=
V2=
V3=