Linear Transformation

Question 1

Another name of Linear Transformation

Answer 1

Linear Map

Question 2

Definition of Linear Transformation

Answer 2

It is mapping from vector space to vector space

Question 3

Notation of Linear Transformation

Answer 3

T: V₁→V₂
Here V₁, V2 are vector spaces

Question 4

T: V₁→V₂
What are V₁, V₂ here

Answer 4

V₁, V₂ are vector spaces
V₁ is the domain of the transformation
V₂ is the codomain of the transformation

Question 5

Can we say linear transformation is mapping from subspace to subspace . Give reason for your answer

Answer 5

Yes we can say linear transformation is mapping from subspace to subspace because subspace is also vector space

Question 6

AX = Y
Convert it into linear transformation notation

Answer 6

A: X→Y

Question 7

A: X→Y
Explain

Answer 7

A is linear transformation matrix which transform the vector X into Vector Y

Question 8

Projection (definition)

Answer 8

T is the projection which projects the higher dimension vector space (vector) into lower vector space (vector)

Question 9

Answer 9

Question 10

Answer 10

Question 11

If A is projection matrix, what it tells about A (2)

Answer 11

A is idempotent and symmetric matrix

Question 12

If vector ‘a’ get doubled then the projection matrix onto ‘a’ __________________

Answer 12

If vector ‘a’ get doubled then the projection matrix onto ‘a’ remains same.

Question 13

Projection matrix onto θ-line

Answer 13

Question 14

Explain column space, null space, row space of projection matrix i.e their relationship

Answer 14

Question 15

General coordinate points of R² in terms of trignometry

Answer 15

(x,y) = (cosθ, sinθ)

Question 16

Transformation matrix
Another names also (2)

Answer 16

Question 17

Rotational matrix of R^2 plane

Answer 17

Question 18

If rotation is clockwise

Answer 18

Question 19

If rotation is clockwise

Answer 19

Question 20

If rotation 2 times with angle theta

Answer 20

Question 21

If rotation 2 times with angle theta

Answer 21

Question 22

If rotation by theta and then by phi

Answer 22

Question 23

Rotaion about x axis

Answer 23

Question 24

Rotaion about x axis

Answer 24

Question 25

Rotation about y axis

Answer 25

Question 26

Rotation about y axis

Answer 26

Question 27

Rotation about z axis

Answer 27

Question 28

Rotation about z axis

Answer 28

Question 29

Reflection matrix about theta line

Answer 29

Question 30

Reflection matrix about theta line

Answer 30

Question 31

Note of reflection matrix

Answer 31

Question 32

To transform R^3 —> R^2

Answer 32

Question 33

To transform R^3 —> R^2

Answer 33

Question 34

To transform R^2 —> R^3

Answer 34

Question 35

Projection matrix onto theta direction in x-y plane

Answer 35

Question 36

Projection onto a plane = ____________

Answer 36

Question 37

Projection onto a plane = ____________

Answer 37

Question 38

Projection to x-y plane

Answer 38

Question 39

Projection to x-y plane

Answer 39