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Let T : R3 --> R3 be defined by T(x, y, z) = (x, y, 0) and S : R2 —> R2 be defined by S(x, y) = (2x, 3y), be linear transformations on the real vector spaces R3 and R2, respectively. Then, which one of the following statement is correct?
  • a)
    T and S are both singular
  • b)
    T and S are both non-singular
  • c)
    T is singular and S is non-singular
  • d)
    S is singular and T is non-singular
Correct answer is option 'C'. Can you explain this answer?
Verified Answer
Let T : R3 --> R3 be defined by T(x, y, z) = (x, y, 0) and S : R2 &...
Let T : R3 -> R3 be the linear transformation defined by

and S : R2 ---> R2 be the linear transformation defined by
S(x, y) = (2x, 3y)
Now,ker T = {(x, y , z) such that T(x,y, z) = 0}
= {(x, y ,z) : (x, y, 0) = (0,0,0)}
= {(0,0,z) : z ∈ R }
therefore, ker T ≠ {0, 0, 0} hence T is singular.
Next,ker S = {(x, y ) : S(x, y) = 0}
= {(x,y) : (2x,3y) = (0, 0)}
= {(0, 0)}
Therefore, S is non-singular.
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Let T : R3 --> R3 be defined by T(x, y, z) = (x, y, 0) and S : R2 &...
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Let T : R3 --> R3 be defined by T(x, y, z) = (x, y, 0) and S : R2 —> R2 be defined by S(x, y) =(2x, 3y), be linear transformations on the real vector spaces R3 and R2, respectively. Then, which one of the following statement is correct?a)T and S are both singularb)T and S are both non-singularc)T is singular and S is non-singulard)S is singular and T is non-singularCorrect answer is option 'C'. Can you explain this answer?
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