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Rank and nullity of linear transformations and matrices

Choose difficulty:

Beginner

Intermediate

Advanced

Let

T:R

² →

R

² be the linear transformation defined by

T(x, y) = (-x, x)

, where

x, y

∈

R

. Then the nullity of

T

is?

a: 0 Explanation

Explanation

b: 1 Explanation

Explanation

c: 2 Explanation

Explanation

d: 4 Explanation

Explanation

Let

T:R

² →

R

² be the linear transformation defined by

T(x, y) = (-x, -y)

, where

x, y

∈

R

. Then the rank of

T

is?

a: 1 Explanation

Explanation

b: 0 Explanation

Explanation

c: 2 Explanation

Explanation

d: 4 Explanation

Explanation

Let

T:R

² →

R

³ be the linear transformation defined by

T(x, y)= (x, x+y, 0)

, where

x, y, z

∈

R

. Then the nullity of

T

is?

a: 0 Explanation

Explanation

b: 1 Explanation

Explanation

c: 2 Explanation

Explanation

d: 3 Explanation

Explanation

Let

T:R

² →

R

³ be the linear transformation defined by

T(x, y) = (-x, -y, 0)

, where

x, y, z

∈

R

. Then the null space of

T

is?

a: {(0, 0,

γ

):

γ

∈

R

} Explanation

Explanation

b: {(

α

,

β

, 0):

α

,

β

∈

R

} Explanation

Explanation

c: {(

α

, 0,

γ

):

α

,

γ

∈

R

} Explanation

Explanation

d: {(0, 0, 0)} Explanation

Explanation

Let

T:R

² →

R

² be the linear transformation defined by

T(x, y)=(x, y)

, where

x, y

∈

R

. Then the range of

T

is?

a: {(

α

,

β

):

α

,

β

∈

R

} Explanation

Explanation

b: {(

α

, 0):

α

∈

R

} Explanation

Explanation

c: {(0,

β

):

β

∈

R

} Explanation

Explanation

d: {(0, 0)} Explanation

Explanation

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