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Let T: R3 ---> R3 be the linear transformation whose matrix with respect to the standard basis of R3 is 
where a, b, c are real numbers not all zero. Then T
where a, b, c are real numbers not all zero. Then T- a)is one-to-one
- b)is onto
- c)does not map any line through the origin onto itself
- d)has rank 1
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
Let T: R3---> R3 be the linear transformation whose matrixwith resp...
Let T : R3 -> R3 be the linear transformation whose matrix with respect to the standard basis of R3 is 
where a, b, c are real number not all zero. The determinant 
= -abc + bac = 0
Thus, Rank of T is 2. Using Rank nullity theorem, Nullity T = 3 - 2 = 1.
Hence, T is not one-one. Also T is not onto because range has two vectors, so it will not generate R3.
Next, we know that the equation of line passing through origin is
Let (l, m, n) be the point on the line. Then (l, m, n) = l(1, 0, 0) + m(0, 1, 0) + n(0 , 0 , 1)
Taking the image under T, we get
T(1,m,n) = T(l(1, 0,0) + m(0,1,0) + n(0,0,1))
= IT(1, 0, 0) + mT(0,1, 0) + nT(0, 0,1)
= l(0, -a, -b) + m(a, 0, -c) + n(b, c, 0)
= (ma + nb, -al + nc, - bl - mc)
But the point (ma + nb, -a l + nc, - bl - mc) does not lie on the line
where a, b, c are real number not all zero. The determinant = -abc + bac = 0Thus, Rank of T is 2. Using Rank nullity theorem, Nullity T = 3 - 2 = 1.
Hence, T is not one-one. Also T is not onto because range has two vectors, so it will not generate R3.
Next, we know that the equation of line passing through origin is
Let (l, m, n) be the point on the line. Then (l, m, n) = l(1, 0, 0) + m(0, 1, 0) + n(0 , 0 , 1)
Taking the image under T, we get
T(1,m,n) = T(l(1, 0,0) + m(0,1,0) + n(0,0,1))
= IT(1, 0, 0) + mT(0,1, 0) + nT(0, 0,1)
= l(0, -a, -b) + m(a, 0, -c) + n(b, c, 0)
= (ma + nb, -al + nc, - bl - mc)
But the point (ma + nb, -a l + nc, - bl - mc) does not lie on the line
Most Upvoted Answer
Let T: R3---> R3 be the linear transformation whose matrixwith resp...
Let T : R3 -> R3 be the linear transformation whose matrix with respect to the standard basis of R3 is 
where a, b, c are real number not all zero. The determinant 
= -abc + bac = 0
Thus, Rank of T is 2. Using Rank nullity theorem, Nullity T = 3 - 2 = 1.
Hence, T is not one-one. Also T is not onto because range has two vectors, so it will not generate R3.
Next, we know that the equation of line passing through origin is
Let (l, m, n) be the point on the line. Then (l, m, n) = l(1, 0, 0) + m(0, 1, 0) + n(0 , 0 , 1)
Taking the image under T, we get
T(1,m,n) = T(l(1, 0,0) + m(0,1,0) + n(0,0,1))
= IT(1, 0, 0) + mT(0,1, 0) + nT(0, 0,1)
= l(0, -a, -b) + m(a, 0, -c) + n(b, c, 0)
= (ma + nb, -al + nc, - bl - mc)
But the point (ma + nb, -a l + nc, - bl - mc) does not lie on the line
where a, b, c are real number not all zero. The determinant = -abc + bac = 0Thus, Rank of T is 2. Using Rank nullity theorem, Nullity T = 3 - 2 = 1.
Hence, T is not one-one. Also T is not onto because range has two vectors, so it will not generate R3.
Next, we know that the equation of line passing through origin is
Let (l, m, n) be the point on the line. Then (l, m, n) = l(1, 0, 0) + m(0, 1, 0) + n(0 , 0 , 1)
Taking the image under T, we get
T(1,m,n) = T(l(1, 0,0) + m(0,1,0) + n(0,0,1))
= IT(1, 0, 0) + mT(0,1, 0) + nT(0, 0,1)
= l(0, -a, -b) + m(a, 0, -c) + n(b, c, 0)
= (ma + nb, -al + nc, - bl - mc)
But the point (ma + nb, -a l + nc, - bl - mc) does not lie on the line
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Let T: R3---> R3 be the linear transformation whose matrixwith respect to the standard basis of R3 iswhere a, b, c are real numbers not all zero. Then Ta)is one-to-oneb)is ontoc)does not map any line through the origin onto itselfd)has rank 1Correct answer is option 'C'. Can you explain this answer?
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Let T: R3---> R3 be the linear transformation whose matrixwith respect to the standard basis of R3 iswhere a, b, c are real numbers not all zero. Then Ta)is one-to-oneb)is ontoc)does not map any line through the origin onto itselfd)has rank 1Correct answer is option 'C'. 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 Let T: R3---> R3 be the linear transformation whose matrixwith respect to the standard basis of R3 iswhere a, b, c are real numbers not all zero. Then Ta)is one-to-oneb)is ontoc)does not map any line through the origin onto itselfd)has rank 1Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let T: R3---> R3 be the linear transformation whose matrixwith respect to the standard basis of R3 iswhere a, b, c are real numbers not all zero. Then Ta)is one-to-oneb)is ontoc)does not map any line through the origin onto itselfd)has rank 1Correct answer is option 'C'. Can you explain this answer?.
Let T: R3---> R3 be the linear transformation whose matrixwith respect to the standard basis of R3 iswhere a, b, c are real numbers not all zero. Then Ta)is one-to-oneb)is ontoc)does not map any line through the origin onto itselfd)has rank 1Correct answer is option 'C'. 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 Let T: R3---> R3 be the linear transformation whose matrixwith respect to the standard basis of R3 iswhere a, b, c are real numbers not all zero. Then Ta)is one-to-oneb)is ontoc)does not map any line through the origin onto itselfd)has rank 1Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let T: R3---> R3 be the linear transformation whose matrixwith respect to the standard basis of R3 iswhere a, b, c are real numbers not all zero. Then Ta)is one-to-oneb)is ontoc)does not map any line through the origin onto itselfd)has rank 1Correct answer is option 'C'. Can you explain this answer?.
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