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Choose the correct matching from A, B, C and D for the transformation T1, T2 and T3 (mappings from R2 to R3) as defined in Group 1 with the statements given in Group 2
Group 1
P.T1x( x, y) = (x , x, 0)
Q.T2(x,y) = (x, x +y ,y)
R.T3(x,y) = (x,x+1,y)
Group 2
1. Linear transformation of rank 2
2. Not a linear transformation
3. Linear transformation of rank 1
Group 1
P.T1x( x, y) = (x , x, 0)
Q.T2(x,y) = (x, x +y ,y)
R.T3(x,y) = (x,x+1,y)
Group 2
1. Linear transformation of rank 2
2. Not a linear transformation
3. Linear transformation of rank 1
- a)P-3, Q-1, R-2
- b)P -1, Q-2, R-3
- c)P - 3, Q - 2, R- 1
- d)P-1, Q-3, R-2
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
Choose the correct matching from A, B, C and D for the transformation ...
Consider the linear transformation defined by
T1(x,y) = (x,x, 0)
Let (x1,x1) and (x2,x2) be two vectors and α, β be scalars. Then
T1[α(x1 y1) + β(x2, y2)]
= T1[(αx1 + βx2, αy1 + βy2)]
= (αx1 + βx2, αx1 + βx2, 0)
= α(x1,x1, 0) + β(x2, x2, 0)
= αT1(x1,y1) + βT1(x2,y2)
or equivalently
Therefore, T1 is linear transformation.
Let (x,y) ∈ ker T1.
Then T(x,y) = (0, 0, 0)
Using the definition of linear transformation T1,we get
(x, x, 0) = (0, 0, 0)
Comparing the components of the co-ordinates, we get,
x = 0 and y is arbitrary
Therefore,ker T1{ = {(0, y ) : y ∈ R}
Hence,dim ker T1 = 1.
Using Rank nullity theorem Rank T1
= 2 - dim ker T1
= 2 - 1 = 1.
Therefore, 'T1 is a linear transformation of rank 1.
Next, consider the linear transformation T2 defined by
T2(x,y) = (x,x+y,y)
Let (x1,y1) and (x2,y2) be any two vectors and α, β are scalars.
Then
Hence, T2 is a linear transformation.
Let (x, y) ∈ ker T.
Then T2(x,y) = (0, 0, 0)
Using the definition of linear transformation T2, we get
(.x, x+ y,y ) = (0, 0, 0)
Comparing the components of the coordinates, we get
x = 0, x + y = 0, y = 0
Solving for x and y, we get x = 0, y = 0
Hence,ker T2 = {(0,0)}.
Therefore,dim ker T2 = 0
Using rank nullity theorem,
Rank T2 = 2 - dim ker T2
= 2 - 0 - 2 .
Hence, T2 is a linear transformation of rank 2.
Next, consider the linear transformation T3 defined by
T3(x,y) = (x,x+1, y)
Then T3(0, 0) =(0, 1, 0)
but (0,1,0) is not a zero of codomain space.
Hence, the image of (0, 0) under T3 is not a zero.
Therefore T3 is not a linear transformation.
T1(x,y) = (x,x, 0)
Let (x1,x1) and (x2,x2) be two vectors and α, β be scalars. Then
T1[α(x1 y1) + β(x2, y2)]
= T1[(αx1 + βx2, αy1 + βy2)]
= (αx1 + βx2, αx1 + βx2, 0)
= α(x1,x1, 0) + β(x2, x2, 0)
= αT1(x1,y1) + βT1(x2,y2)
or equivalently
Therefore, T1 is linear transformation.
Let (x,y) ∈ ker T1.
Then T(x,y) = (0, 0, 0)
Using the definition of linear transformation T1,we get
(x, x, 0) = (0, 0, 0)
Comparing the components of the co-ordinates, we get,
x = 0 and y is arbitrary
Therefore,ker T1{ = {(0, y ) : y ∈ R}
Hence,dim ker T1 = 1.
Using Rank nullity theorem Rank T1
= 2 - dim ker T1
= 2 - 1 = 1.
Therefore, 'T1 is a linear transformation of rank 1.
Next, consider the linear transformation T2 defined by
T2(x,y) = (x,x+y,y)
Let (x1,y1) and (x2,y2) be any two vectors and α, β are scalars.
Then
Hence, T2 is a linear transformation.Let (x, y) ∈ ker T.
Then T2(x,y) = (0, 0, 0)
Using the definition of linear transformation T2, we get
(.x, x+ y,y ) = (0, 0, 0)
Comparing the components of the coordinates, we get
x = 0, x + y = 0, y = 0
Solving for x and y, we get x = 0, y = 0
Hence,ker T2 = {(0,0)}.
Therefore,dim ker T2 = 0
Using rank nullity theorem,
Rank T2 = 2 - dim ker T2
= 2 - 0 - 2 .
Hence, T2 is a linear transformation of rank 2.
Next, consider the linear transformation T3 defined by
T3(x,y) = (x,x+1, y)
Then T3(0, 0) =(0, 1, 0)
but (0,1,0) is not a zero of codomain space.
Hence, the image of (0, 0) under T3 is not a zero.
Therefore T3 is not a linear transformation.
Most Upvoted Answer
Choose the correct matching from A, B, C and D for the transformation ...
Explanation:
To determine if a transformation is linear, we need to check if it satisfies the two properties of linearity:
1. Additivity: T(u + v) = T(u) + T(v)
2. Homogeneity: T(cu) = cT(u)
Transformation T1:
- T1(x, y) = (x, x, 0)
Additivity:
- T1(u + v) = (u + v, u + v, 0)
- T1(u) + T1(v) = (u, u, 0) + (v, v, 0) = (u + v, u + v, 0)
Additivity property is satisfied.
Homogeneity:
- T1(cu) = (cu, cu, 0)
- cT1(u) = c(u, u, 0) = (cu, cu, 0)
Homogeneity property is satisfied.
Therefore, T1 is a linear transformation.
Transformation T2:
- T2(x, y) = (x, x + y, y)
Additivity:
- T2(u + v) = (u + v, u + v + (v + w), v + w)
- T2(u) + T2(v) = (u, u + v, v) + (v, v + w, w) = (u + v, 2u + 2v + w, v + w)
Additivity property is not satisfied since T2(u + v) ≠ T2(u) + T2(v).
Therefore, T2 is not a linear transformation.
Transformation T3:
- T3(x, y) = (x, x + 1, y)
Additivity:
- T3(u + v) = (u + v, u + v + 1, v + w)
- T3(u) + T3(v) = (u, u + 1, v) + (v, v + 1, w) = (u + v, 2u + 2v + 1, v + w)
Additivity property is not satisfied since T3(u + v) ≠ T3(u) + T3(v).
Therefore, T3 is not a linear transformation.
Rank of the transformations:
- The rank of a linear transformation is the dimension of the image space.
- For T1, the image space is a subspace of R3 spanned by the vectors (1, 1, 0). The dimension of this subspace is 1, so the rank of T1 is 1.
- For T2, the image space is a subspace of R3 spanned by the vectors (1, 0, 0) and (0, 1, 1). The dimension of this subspace is 2, so the rank of T2 is 2.
- For T3, the image space is a subspace of R3 spanned by the vectors (1, 1, 0) and (0, 1, 1). The
To determine if a transformation is linear, we need to check if it satisfies the two properties of linearity:
1. Additivity: T(u + v) = T(u) + T(v)
2. Homogeneity: T(cu) = cT(u)
Transformation T1:
- T1(x, y) = (x, x, 0)
Additivity:
- T1(u + v) = (u + v, u + v, 0)
- T1(u) + T1(v) = (u, u, 0) + (v, v, 0) = (u + v, u + v, 0)
Additivity property is satisfied.
Homogeneity:
- T1(cu) = (cu, cu, 0)
- cT1(u) = c(u, u, 0) = (cu, cu, 0)
Homogeneity property is satisfied.
Therefore, T1 is a linear transformation.
Transformation T2:
- T2(x, y) = (x, x + y, y)
Additivity:
- T2(u + v) = (u + v, u + v + (v + w), v + w)
- T2(u) + T2(v) = (u, u + v, v) + (v, v + w, w) = (u + v, 2u + 2v + w, v + w)
Additivity property is not satisfied since T2(u + v) ≠ T2(u) + T2(v).
Therefore, T2 is not a linear transformation.
Transformation T3:
- T3(x, y) = (x, x + 1, y)
Additivity:
- T3(u + v) = (u + v, u + v + 1, v + w)
- T3(u) + T3(v) = (u, u + 1, v) + (v, v + 1, w) = (u + v, 2u + 2v + 1, v + w)
Additivity property is not satisfied since T3(u + v) ≠ T3(u) + T3(v).
Therefore, T3 is not a linear transformation.
Rank of the transformations:
- The rank of a linear transformation is the dimension of the image space.
- For T1, the image space is a subspace of R3 spanned by the vectors (1, 1, 0). The dimension of this subspace is 1, so the rank of T1 is 1.
- For T2, the image space is a subspace of R3 spanned by the vectors (1, 0, 0) and (0, 1, 1). The dimension of this subspace is 2, so the rank of T2 is 2.
- For T3, the image space is a subspace of R3 spanned by the vectors (1, 1, 0) and (0, 1, 1). The
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Choose the correct matching from A, B, C and D for the transformation T1, T2and T3 (mappings from R2 to R3) as defined in Group 1 with the statements given in Group 2Group 1P.T1x( x, y) = (x , x, 0)Q.T2(x,y) = (x, x +y ,y)R.T3(x,y) = (x,x+1,y)Group 21. Linear transformation of rank 22. Not a linear transformation3. Linear transformation of rank 1a)P-3, Q-1, R-2b)P -1, Q-2, R-3c)P - 3, Q - 2, R- 1d)P-1, Q-3, R-2Correct answer is option 'A'. Can you explain this answer?
Question Description
Choose the correct matching from A, B, C and D for the transformation T1, T2and T3 (mappings from R2 to R3) as defined in Group 1 with the statements given in Group 2Group 1P.T1x( x, y) = (x , x, 0)Q.T2(x,y) = (x, x +y ,y)R.T3(x,y) = (x,x+1,y)Group 21. Linear transformation of rank 22. Not a linear transformation3. Linear transformation of rank 1a)P-3, Q-1, R-2b)P -1, Q-2, R-3c)P - 3, Q - 2, R- 1d)P-1, Q-3, R-2Correct answer is option 'A'. Can you explain this answer? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Choose the correct matching from A, B, C and D for the transformation T1, T2and T3 (mappings from R2 to R3) as defined in Group 1 with the statements given in Group 2Group 1P.T1x( x, y) = (x , x, 0)Q.T2(x,y) = (x, x +y ,y)R.T3(x,y) = (x,x+1,y)Group 21. Linear transformation of rank 22. Not a linear transformation3. Linear transformation of rank 1a)P-3, Q-1, R-2b)P -1, Q-2, R-3c)P - 3, Q - 2, R- 1d)P-1, Q-3, R-2Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Choose the correct matching from A, B, C and D for the transformation T1, T2and T3 (mappings from R2 to R3) as defined in Group 1 with the statements given in Group 2Group 1P.T1x( x, y) = (x , x, 0)Q.T2(x,y) = (x, x +y ,y)R.T3(x,y) = (x,x+1,y)Group 21. Linear transformation of rank 22. Not a linear transformation3. Linear transformation of rank 1a)P-3, Q-1, R-2b)P -1, Q-2, R-3c)P - 3, Q - 2, R- 1d)P-1, Q-3, R-2Correct answer is option 'A'. Can you explain this answer?.
Choose the correct matching from A, B, C and D for the transformation T1, T2and T3 (mappings from R2 to R3) as defined in Group 1 with the statements given in Group 2Group 1P.T1x( x, y) = (x , x, 0)Q.T2(x,y) = (x, x +y ,y)R.T3(x,y) = (x,x+1,y)Group 21. Linear transformation of rank 22. Not a linear transformation3. Linear transformation of rank 1a)P-3, Q-1, R-2b)P -1, Q-2, R-3c)P - 3, Q - 2, R- 1d)P-1, Q-3, R-2Correct answer is option 'A'. Can you explain this answer? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Choose the correct matching from A, B, C and D for the transformation T1, T2and T3 (mappings from R2 to R3) as defined in Group 1 with the statements given in Group 2Group 1P.T1x( x, y) = (x , x, 0)Q.T2(x,y) = (x, x +y ,y)R.T3(x,y) = (x,x+1,y)Group 21. Linear transformation of rank 22. Not a linear transformation3. Linear transformation of rank 1a)P-3, Q-1, R-2b)P -1, Q-2, R-3c)P - 3, Q - 2, R- 1d)P-1, Q-3, R-2Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Choose the correct matching from A, B, C and D for the transformation T1, T2and T3 (mappings from R2 to R3) as defined in Group 1 with the statements given in Group 2Group 1P.T1x( x, y) = (x , x, 0)Q.T2(x,y) = (x, x +y ,y)R.T3(x,y) = (x,x+1,y)Group 21. Linear transformation of rank 22. Not a linear transformation3. Linear transformation of rank 1a)P-3, Q-1, R-2b)P -1, Q-2, R-3c)P - 3, Q - 2, R- 1d)P-1, Q-3, R-2Correct answer is option 'A'. Can you explain this answer?.
Solutions for Choose the correct matching from A, B, C and D for the transformation T1, T2and T3 (mappings from R2 to R3) as defined in Group 1 with the statements given in Group 2Group 1P.T1x( x, y) = (x , x, 0)Q.T2(x,y) = (x, x +y ,y)R.T3(x,y) = (x,x+1,y)Group 21. Linear transformation of rank 22. Not a linear transformation3. Linear transformation of rank 1a)P-3, Q-1, R-2b)P -1, Q-2, R-3c)P - 3, Q - 2, R- 1d)P-1, Q-3, R-2Correct answer is option 'A'. Can you explain this answer? in English & in Hindi are available as part of our courses for Mathematics.
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Choose the correct matching from A, B, C and D for the transformation T1, T2and T3 (mappings from R2 to R3) as defined in Group 1 with the statements given in Group 2Group 1P.T1x( x, y) = (x , x, 0)Q.T2(x,y) = (x, x +y ,y)R.T3(x,y) = (x,x+1,y)Group 21. Linear transformation of rank 22. Not a linear transformation3. Linear transformation of rank 1a)P-3, Q-1, R-2b)P -1, Q-2, R-3c)P - 3, Q - 2, R- 1d)P-1, Q-3, R-2Correct answer is option 'A'. Can you explain this answer?, a detailed solution for Choose the correct matching from A, B, C and D for the transformation T1, T2and T3 (mappings from R2 to R3) as defined in Group 1 with the statements given in Group 2Group 1P.T1x( x, y) = (x , x, 0)Q.T2(x,y) = (x, x +y ,y)R.T3(x,y) = (x,x+1,y)Group 21. Linear transformation of rank 22. Not a linear transformation3. Linear transformation of rank 1a)P-3, Q-1, R-2b)P -1, Q-2, R-3c)P - 3, Q - 2, R- 1d)P-1, Q-3, R-2Correct answer is option 'A'. Can you explain this answer? has been provided alongside types of Choose the correct matching from A, B, C and D for the transformation T1, T2and T3 (mappings from R2 to R3) as defined in Group 1 with the statements given in Group 2Group 1P.T1x( x, y) = (x , x, 0)Q.T2(x,y) = (x, x +y ,y)R.T3(x,y) = (x,x+1,y)Group 21. Linear transformation of rank 22. Not a linear transformation3. Linear transformation of rank 1a)P-3, Q-1, R-2b)P -1, Q-2, R-3c)P - 3, Q - 2, R- 1d)P-1, Q-3, R-2Correct answer is option 'A'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice Choose the correct matching from A, B, C and D for the transformation T1, T2and T3 (mappings from R2 to R3) as defined in Group 1 with the statements given in Group 2Group 1P.T1x( x, y) = (x , x, 0)Q.T2(x,y) = (x, x +y ,y)R.T3(x,y) = (x,x+1,y)Group 21. Linear transformation of rank 22. Not a linear transformation3. Linear transformation of rank 1a)P-3, Q-1, R-2b)P -1, Q-2, R-3c)P - 3, Q - 2, R- 1d)P-1, Q-3, R-2Correct answer is option 'A'. Can you explain this answer? tests, examples and also practice Mathematics tests.
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