Exam 2

Question 1

Testing for row equivalence

Answer 1

Every matrix row equivalence class reduces to the same rref matrix.

Question 2

Showing row operations to relate two matrices as row-equivalent

Answer 2

A = (α1, α2…)

Show that B = (β1=α1, β2=2α2… etc)

Question 3

Echelon form definition

Answer 3

No non-zero row is a linear combination of any of the other rows.
Verify:
row1 /= (row2)c1 + (row3)c2 …+ (rowN)cN

Question 4

A system of equations has ___ matrices associated with it.

Answer 4

3:
Coefficient matrix
Homogeneous matrix
Non-homogeneous matrix

Question 5

Equivalent statements for:

A is nonsingular

Answer 5

A is nonsingular.
rref [A] = identity matrix.
The homogeneous system has one solution: the zero vector.

Question 6

T/F: All nonsingular matrices are in the same row-equivalence class.

Answer 6

True.

Nonsingular means rref[A]= identity matrix

Question 7

T/F: All NxN (square) matrices are in the same row equivalence class.

Answer 7

False

Question 8

If two systems (“systems” includes “=d”, nonhomogeneous) have row equivalent matrices, do they have the same solution?

Answer 8

Yes

Question 9

How many members can there be in a row equivalence class?

Answer 9

Either one (the zero matrix) or infinite.

Question 10

Rules to determine if a set is a vector space.

Answer 10

Addition: u+v ε V
Scalar multiplication: rv ε V
Zero vector: 0 ε V

Question 11

The improper subspaces of V

Answer 11

V itself and the trivial subspace (0)

Question 12

Constructing a subset

Answer 12

Put some restriction on the vector space.

I.e. “such that…”

Question 13

Constructing a subset that is a subspace of the vector space.

Answer 13

Put a restriction on the vector space “such that…” which abides by the rules of a space (addition, scalar multiplication, zero vector).

Question 14

Equivalent statements:

S is a subspace of V

Answer 14

S is a subspace of V.

S is “closed” (valid) under linear combinations of any N of its vectors.

Question 15

S is a subspace of V. Find all vectors that can be used in equivalent statements.

Answer 15

Parameterize the statement (free variables times vectors) so that S is the set of all linear combinations of those vectors

Question 16

What is a linear combination?

Answer 16

Coefficients times the vectors of a system

Question 17

What is span?

Answer 17

The span of a subset S is:
the set of all linear combinations of vectors within S

Note: [S] is a vector space, and will be in the form of a line (one parameter), plane (two parameters) etc

Question 18

Test for linear independence

Answer 18

Set S is linearly independent iff
c1s1 + … + cNsN = 0
has ONLY the trivial solution:
c1 = c2 = … = cN = 0

Note: S is linearly dependent if the restriction is c1 = c2 = etc… Or if the zero vector is included in the set

Question 19

How to identify/remove the unnecessary vectors in a linearly dependent set? (How to create a subset with the same span)

Answer 19

Vectors times coefficients (c1, c2, etc) = 0.
Convert to a system or matrix, solve, discard any of the original vectors that are attached to the resulting free variables

Question 20

Minimal =

Answer 20

Linearly independent

Question 21

Maximal =

Answer 21

Spans the space

Question 22

Linearly dependent subset of a linearly dependent set S:

Answer 22

Does not exist

Question 23

Linearly independent subset of a linearly independent set S:

Answer 23

S, the original set itself

Question 24

Define: basis

Answer 24

A minimal and maximal set

linearly independent and spans the space

Question 25

Does it span the space?

Answer 25

= “Do c1, c2, etc… exist such that you can produce all the vectors in the space?”

For a matrix: transpose, reduce to rref, number of nonzero rows must match # of dimensions in a space if it spans (bc that’s the # of linearly independent vectors in the set)

Question 26

“Representation of vector v with respect to basis B”

RepBv =

Answer 26

RepBv =
( c1 )
( c2 )
( ...  )
( cN )

Where v = c1b1 +…+ cNbN
(b’s = members of basis B)

Question 27

Find span(S) in V.

Answer 27

Parameterize the set (c1, c2, etc) and
set equal to the span of V
(like (x, y, z) or (a0 + a1x + a2x^2),
Put into a system of eq’s and solve. Right side of eq’s are either unrestrained (then [S]=V) or restrained by some combo of parameters (a’s or xyz’s). Put parameters with the span of V (like R3 vector or P2 polynomial) to get the span of S