To show: The set { T(x_1), ... ,T(x_k)} is a linearly independent subset of R^{m} Given: Let T : T : R^{n} rightarrow R^{m} be a linear transformation with nulity zero. If S={x_{1}, cdots , x_{k}} is a linearly independent subset of R^{n}.

To show:
The set is a linearly independent subset of ${R}^{m}$
Given:
Let be a linear transformation with nulity zero. If is a linearly independent subset of ${R}^{n}.$
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hosentak
Calculation:
Consider
Since, T is a linear transformation. So,

Use the result of equation (1) in the above equation.

Use the results of equation (2), in the above equation.

Now, is a linearly independent subset of ${R}^{n}$
This implies any linear combination of the elements of S will give the value of constants as 0.
This means
Therefore, is a linearly independent subset of ${R}^{m}$
Conclusion:
Hence, is linearly independent subset of ${R}^{m}$