Lesson test: Rank, null space and solution sets

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Rank, null space and solution sets

1.[2p]
The column space of $A$ is

the set of all vectors $A 𝐱$ , that is, everything the transformation can reach

Correct

the set of all $𝐱$ with $A 𝐱 = 𝟎$

The answer is: the set of all vectors $A 𝐱$ , that is, everything the transformation can reach

the span of the rows of $A$

The answer is: the set of all vectors $A 𝐱$ , that is, everything the transformation can reach

the set of solutions of $A 𝐱 = 𝐛$

The answer is: the set of all vectors $A 𝐱$ , that is, everything the transformation can reach

2.[2p]
A matrix has eight columns and rank three. What is the dimension of its null space?

CorrectNot quite: 5

3.[2p]
What is the rank of $[(1, 1), (2, 2), (3, 3)]$ ?

CorrectNot quite: 1

4.[2p]
The solution set of $A 𝐱 = 𝐛$ is a subspace whenever a solution exists.

True

The answer is: False

False

Correct

5.[2p]
The complete solution of a consistent system is

one particular solution plus the whole null space

Correct

the null space alone

The answer is: one particular solution plus the whole null space

the column space shifted by $𝐛$

The answer is: one particular solution plus the whole null space

always a single point

The answer is: one particular solution plus the whole null space

6.[3p]
For the matrix $A = [(1, 2, 3), (2, 4, 6), (0, 1, 1)]$ , which statements are true?

Select all that apply

Its null space is the line spanned by $(- 2, 1, 0)$

Correct

Its rank is 2

Correct

Its column space is a plane in $R^{3}$

Correct

It is invertible

The answer is: Its null space is the line spanned by $(- 2, 1, 0)$ , Its rank is 2, Its column space is a plane in $R^{3}$

7.[3p]
Why is the null space perpendicular to the row space?

Each component of $A 𝐱$ is a row dotted with $𝐱$ , and all of them are zero

Correct

Because both are subspaces of the same dimension

The answer is: Each component of $A 𝐱$ is a row dotted with $𝐱$ , and all of them are zero

Because elimination makes the rows orthogonal

The answer is: Each component of $A 𝐱$ is a row dotted with $𝐱$ , and all of them are zero

Because the determinant is zero

The answer is: Each component of $A 𝐱$ is a row dotted with $𝐱$ , and all of them are zero

8.[3p]
For an $m$ by $n$ matrix of rank $r$ , match each subspace to its dimension.

Column space
Null space
Row space
Null space of the transpose

n minus r
r
r
m minus r

Show the answer

Column space: r Null space: n minus r Row space: r Null space of the transpose: m minus r

9.[2p]
For a square matrix, what must the nullity be for $A 𝐱 = 𝐛$ to have exactly one solution?

CorrectNot quite: 0

1.[2p] The column space of A is

2.[2p] A matrix has eight columns and rank three. What is the dimension of its null space?

3.[2p] What is the rank of [(1,1),(2,2),(3,3)]?

4.[2p] The solution set of A𝐱=𝐛 is a subspace whenever a solution exists.

5.[2p] The complete solution of a consistent system is

6.[3p] For the matrix A=[(1,2,3),(2,4,6),(0,1,1)], which statements are true?

7.[3p] Why is the null space perpendicular to the row space?

8.[3p] For an m by n matrix of rank r, match each subspace to its dimension.

9.[2p] For a square matrix, what must the nullity be for A𝐱=𝐛 to have exactly one solution?

1.[2p]
The column space of $A$ is

2.[2p]
A matrix has eight columns and rank three. What is the dimension of its null space?

3.[2p]
What is the rank of $[(1, 1), (2, 2), (3, 3)]$ ?

4.[2p]
The solution set of $A 𝐱 = 𝐛$ is a subspace whenever a solution exists.

5.[2p]
The complete solution of a consistent system is

6.[3p]
For the matrix $A = [(1, 2, 3), (2, 4, 6), (0, 1, 1)]$ , which statements are true?

7.[3p]
Why is the null space perpendicular to the row space?

8.[3p]
For an $m$ by $n$ matrix of rank $r$ , match each subspace to its dimension.

9.[2p]
For a square matrix, what must the nullity be for $A 𝐱 = 𝐛$ to have exactly one solution?