Solved: "Prove the relation T=S."

CoormaBak9

Answered question

2020-12-31

Prove the relation

T = S .

Bertha Stark

Skilled2021-01-01Added 96 answers

The vector space

B = {v_{1}, \dots, v_{k}} is a basis for vector space V, and S : V \to W, T : V \to W are linear transformations such that S (v_{i}) = T (v_{i}) .

Suppose vector v in V.

v = a_{1} v_{1} + a_{2} v_{2} + \dots + a_{n} v_{n}

where,

a_{1}, a_{2}, \dots a_{n}

are scalars.
Suppose,

S (v) = S (a_{1} v_{1} + a_{2} v_{2} + \dots + a_{n} v_{n}) . (1)

As S is a linear transformation therefore,

S (v_{1} + v_{2}) = S (v_{1}) + S (v_{2})

S (a v) = a S (v)

S (v) = S (a_{1} v_{1} + a_{2} v_{2} + \dots + a_{n} v_{n})

= S (a_{1} v_{1}) + S (a_{2} v_{2}) + \dots + S (a_{n} v_{n})

= a_{1} S (v_{1}) + a_{2} S (v_{2}) + \dots + a_{n} S (v_{n})

Also,

S (v_{i}) = T (v_{i})

So,

S (v) = a_{1} S (v_{1}) + a_{2} S (v_{2}) + \dots + a_{n} S (v_{n})

= a_{1} T (v_{1}) + a_{2} T (v_{2}) + \dots + a_{n} T (v_{n})

= T (a_{1} v_{1} + a_{2} v_{2} + \dots + a_{n} v_{n})

= T (v)

As, v in V is arbitrary.

S (v) = T (v)

for all v in V.
This proves that

S = T .

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