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Let W be a vector space over R and let T : R6 → W be a linear transformation w.r.t., S = {Te2, Te4, Te6} spans W., Which one of the following must be true?
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Let W be a vector space over R and let T : R6 → W be a linear transfor...
Answer:
Statement: The linear transformation T : R6 → W is onto.
Explanation:
To prove that the linear transformation T : R6 → W is onto, we need to show that for every vector w in W, there exists a vector x in R6 such that T(x) = w.
Step 1: Show that {Te2, Te4, Te6} spans W
Since {Te2, Te4, Te6} spans W, every vector in W can be written as a linear combination of Te2, Te4, and Te6. Therefore, for any vector w in W, we can write:
w = a1(Te2) + a2(Te4) + a3(Te6)
where a1, a2, and a3 are scalars.
Step 2: Express w in terms of a vector x in R6
Since T is a linear transformation, we can write:
T(x) = T(a1e2 + a2e4 + a3e6)
By linearity, we have:
T(x) = a1T(e2) + a2T(e4) + a3T(e6)
T(x) = a1(Te2) + a2(Te4) + a3(Te6)
Comparing this with the expression for w, we can see that T(x) = w.
Step 3: Conclusion
Since we have shown that for every vector w in W, there exists a vector x in R6 such that T(x) = w, we can conclude that the linear transformation T : R6 → W is onto.
Statement: The linear transformation T : R6 → W is onto.
Explanation:
To prove that the linear transformation T : R6 → W is onto, we need to show that for every vector w in W, there exists a vector x in R6 such that T(x) = w.
Step 1: Show that {Te2, Te4, Te6} spans W
Since {Te2, Te4, Te6} spans W, every vector in W can be written as a linear combination of Te2, Te4, and Te6. Therefore, for any vector w in W, we can write:
w = a1(Te2) + a2(Te4) + a3(Te6)
where a1, a2, and a3 are scalars.
Step 2: Express w in terms of a vector x in R6
Since T is a linear transformation, we can write:
T(x) = T(a1e2 + a2e4 + a3e6)
By linearity, we have:
T(x) = a1T(e2) + a2T(e4) + a3T(e6)
T(x) = a1(Te2) + a2(Te4) + a3(Te6)
Comparing this with the expression for w, we can see that T(x) = w.
Step 3: Conclusion
Since we have shown that for every vector w in W, there exists a vector x in R6 such that T(x) = w, we can conclude that the linear transformation T : R6 → W is onto.
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Let W be a vector space over R and let T : R6 → W be a linear transformation w.r.t., S = {Te2, Te4, Te6} spans W., Which one of the following must be true?
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