Chapter 4

Question 1

Traits of Subspace

Answer 1

1. Contains zero vector
2. Closed under addition
3. Closed under multiplication
4. Determinant Zero

Question 2

Three ways to determine Subspace

Answer 2

1. Check if System Consistent
2. Check if determinant 0
3. Check traits of subspace

Question 3

Find a specific vector u in V and a specific scalar c such
that cu is not in V -
Steps

Answer 3

1. Create any U/V vector
2. Scale it by anything that will break condition

Question 4

Find vectors u and v such that
W = Span {u, v}

Answer 4

1. Break given matrix into separate vectors (u, v)

Question 5

Determine if W in Nul(A)

Answer 5

1. Multiply W and A
2. Multiplied vector must be zero matrix for True

Question 6

Find Span set for Nul(A)

Answer 6

1. Augment matrix by zero vector
2. Row reduce all the way
3. List non-pivot/free variable vectors after x1 =, x2=, x3= etc

Question 7

Find A such that given set is Col(A)

Answer 7

Split into vectors based off the variable

Question 8

Find K such that Nul(A) is subspace

Answer 8

Number of Columns

Question 9

Find L such that Col(A) is subspace

Answer 9

Number of Rows

Question 10

P2 is a subspace of P3

Answer 10

TRUE

Question 11

R2 is a subspace of R3

Answer 11

FALSE

Question 12

When given invertible upper-triangular matrix, what is noticeable about diagonal elements?

Answer 12

Diagonal elements are Non-Zero

Question 13

Find Nonzero vector in Nul(A)

Answer 13

1. Augment A by zero vector
2. Row Reduce All the way
3. Do x1=,x2=, thing etc.
4. Choose free variable

Question 14

Find nonzero vector in Col(A)

Answer 14

Choose any column

Question 15

Find nonzero vector in Row(A)

Answer 15

1. Transpose A
2. Choose any column

Question 16

Determine if W in Col(A)

Answer 16

1. Augment A with 0 Vector
2. Row Reduce All the way
3. Check if consistent system

Question 17

How to find Normal Basis

Answer 17

1. Row Reduce all the way
2. Pivot columns are Basis

Question 18

How to find Nullspace Basis

Answer 18

1. Augment A by 0 vector
2. Row Reduce all the way
3. Free variable/Span (x1=, x2=, etc) is Basis

Question 19

Find x

Answer 19

1. [x]b are scalars.
2. Multiply vectors given by the scalars
3. Add the vectors

Question 20

Find [x]b

Answer 20

1. Augment matrix by X
2. Row reduce all the way
3. Rightside of matrix is result

Question 21

Find coordinate of changes

Answer 21

1. Simply put given vectors together into matrix

Question 22

What is Rank

Answer 22

Number of pivots

Question 23

What is Nullity

Answer 23

Number of free variable columns