Answer :
To determine which of the given statements are true, let's analyze the properties of the linear transformation T represented by the standard matrix:
0 2
-1 3
(i) T maps R^3 onto R:
For T to map R^3 onto R, every element in R must have a pre-image in R^3 under T. In this case, since the second column of the matrix contains nonzero entries, we can conclude that T maps R^3 onto R. Therefore, statement (i) is true.
(ii) T maps R onto R^3:
For T to map R onto R^3, every element in R^3 must have a pre-image in R under T. Since the matrix does not have a third column, we cannot conclude that every element in R^3 has a pre-image in R. Therefore, statement (ii) is false.
(iii) T is onto:
A linear transformation T is onto if and only if its range equals the codomain. In this case, since the second column of the matrix is nonzero, the range of T is all of R. Therefore, T is onto. Statement (iii) is true.
(iv) T is one-to-one:
A linear transformation T is one-to-one if and only if its null space contains only the zero vector. To determine this, we can find the null space of the matrix. Solving the equation T(x) = 0, we get:
0x + 2y - z = 0
-x + 3*y = 0
From the second equation, we can express x in terms of y: x = 3y. Substituting this into the first equation, we get:
0 + 2y - z = 0
2y = z
This implies that z must be a multiple of 2y. Therefore, the null space of T contains nonzero vectors, indicating that T is not one-to-one. Statement (iv) is false.
Based on the analysis above, the correct answer is:
A. (i) and (iii) only.
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