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}.

Question

To show:
The set

{T (x_{1}), \dots, T (x_{k})}

is a linearly independent subset of

R^{m}

Given:
Let

T : T : R^{n} \to R^{m}

be a linear transformation with nulity zero. If

S = {x_{1}, \dots, x_{k}}

is a linearly independent subset of

R^{n} .

Answer 1

Calculation:
Consider

a_{1} T (x_{1}) + \dots a_{k} T (x_{k}) = θ_{m}

Since, T is a linear transformation. So,

T (a_{1} x_{1}) +, \dots + T (a_{k} x_{k}) = θ_{m}

T (a_{1} x_{1}) +, \dots + (a_{k} x_{k}) = θ_{m}

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

T (a_{1} x_{1} + \dots + a_{k} x_{k}) = T (θ_{n})

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

a_{1} x_{1} + \dots + a_{k} x_{k} = θ_{n}

Now,

S = {x_{1}, \dots x_{k}}

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

a_{1} = 0, \dots, a_{k} = 0

Therefore,

{T (x_{1}), \dots, T (x_{k})}

is a linearly independent subset of

R^{m}

Conclusion:
Hence,

{T (x_{1}), \dots, T (x_{k})}

is linearly independent subset of

R^{m}

Expert Answer