4.4 - 4.6 review

Axes, Axes, Axes!

100

Define the standard basis for R^3.

Define a basis B for R^3 that is not the standard basis.

Standard basis {e1, e2, e3}

200

Find the vector x determined by the given coordinate vector [x]_B and the given basis B:

B= [[3],[5]], [[4],[6]] and vec [x]_B =[[5],[3]]

[[27],[43]]

300

Find the coordinate vector [x]_B of x relative to the given basis:

vec b_1=[[1],[1],[3]], vec b_2=[[2],[1],[8]], vec b_3=[[1],[-1],[3]] 

and vecx=[[0],[0],[2]]

[x]_B =[[-3/2],[1],[-1/2]]

400

"Determine whether the sets" 

"of polynomials form a basis for" P_3

3+7t, 5+t-2t^3,t-2t^2, 1+16t-6t^2+2t^3 

no

The n-th Dimension

100

All bases for vector space W have what in common?

They all have the same number of vectors.

200

Find a basis for the subspace. State the dimension:

[[p-2q],[2p+5r],[-2q+2r],[-3p+6r]] " such that " p,q,r in RR

[[1],[2],[0],[-3]],[[-2],[0],[-2],[0]],[[0],[5],[2],[6]] " dimension = 3"

300

Determine the dimension of the nulspace for the following matrix A:

[[1, -6, 9, 0, -2],[0, 1, 2, -4, 5],[0, 0, 0, 5, 1],[0, 0, 0, 0, 0]]

Nullity = 2

400

Determine the dimension of the column space for the following matrix A:

[[1, -6, 9, 0, -2],[0, 1, 2, -4, 5],[0, 0, 0, 5, 1],[0, 0, 0, 0, 0]]

dim col (a) = 3

Lieutenant Sergeant Major

100

What is rank?

The dimension of the column space of a matrix.

200

What is the Rank Theorem?

For an m by n matrix A, 

rank A + dim nul A = n

300

if a 4x7 matrix has rank 3, find dim nul A, dim row A, and the rank of A transpose

dimnulA=4

dim row A = 3

rank of A^T=3

400

2

Math Potporri

100

How many possible March Madness brackets exist each year?

2^63 " or " 9.223372" quintillion"

200

What is Mr. Laviolette's favorite Transcendental Number?

e

300

int_{-1}^{1}sin(x)/(5+x^2)dx

0

(This function is odd)

400

Prove whether or not the following graphs are isomorphic.

Not Isomorphic.