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Prove that the zero transformation and the identity transformation are linear transformations. (a) the zero transformation Let V and W be vector spaces, let v and w be vectors in V, let c be a scalar, and let T: VW be the zero transformation. Which of the following proves the zero transformation from V to W is a linear transformation? (Select all that apply.) OT(vw) = 0 = 0·0= T(v) T(w) ⒸT(cv) = 0 = co= CT(v) OT(v + w)=0=0+0= T(v) + T(w) ⒸT(cvw) = 0 = coo=cT(v) T(w) OCT(v + w) = c(0+ 0) = CT(v) + cT(w) OT(c + v) = c+0=c+T(v) (b) the identity transformation Let V be a vector space, let v and w be vectors in V, let e be a scalar, and let T: VV be the identity transformation. Which of the following proves the identity transformation from V to V is a linear transformation? (Select all that apply.) OT(v + w) = v+w= T(v) + T(w) T(cv) cv=cT(v) T(cvw) = cvw=cT(v) T(w) T(c + v) = c+v=c+T(v) OCT(v + w) = c(v + w) = CT(v) + cT(w) OT(vw) = vw = T(v) T(w)

Prove that the zero transformation and the identity transformation are linear transformations. (a) the zero transformation Let V and W be vector spaces, let v and w be vectors in V, let c be a scalar, and let T: VW be the zero transformation. Which of the following proves the zero transformation from V to W is a linear transformation? (Select all that apply.) OT(vw) = 0 = 0·0= T(v) T(w) ⒸT(cv) = 0 = co= CT(v) OT(v + w)=0=0+0= T(v) + T(w) ⒸT(cvw) = 0 = coo=cT(v) T(w) OCT(v + w) = c(0+ 0) = CT(v) + cT(w) OT(c + v) = c+0=c+T(v) (b) the identity transformation Let V be a vector space, let v and w be vectors in V, let e be a scalar, and let T: VV be the identity transformation. Which of the following proves the identity transformation from V to V is a linear transformation? (Select all that apply.) OT(v + w) = v+w= T(v) + T(w) T(cv) cv=cT(v) T(cvw) = cvw=cT(v) T(w) T(c + v) = c+v=c+T(v) OCT(v + w) = c(v + w) = CT(v) + cT(w) OT(vw) = vw = T(v) T(w)

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Prove that the zero transformation and the identity transformation are linear transformations. (a) the zero transformation Let V and W be vector spaces, let v and w be vectors in V, let c be a scalar, and let T: V→ W be the zero transformation. Which of the following proves the zero transformation from V to W is a linear transformation? (Select all that apply.) T(vw) = 0 = 0·0= T(v) T(w) T(cv) = 0 = c0 = cT(v) T(v + w) = 0 = 0 + 0 = T(v) + T(w) T(cvw) = 0 = c0 0 = cT(v) T(w) OCT(v + w) = c(0 + 0) = cT(v) + CT(w) OT(c + v) = c + 0 = c + T(v) (b) the identity transformation Let V be a vector space, let v and w be vectors in V, let c be a scalar, and let T: V→ V be the identity transformation. Which of the following proves the identity transformation from V to V is a linear transformation? (Select all that apply.) T(v + w) = v + w = T(v) + T(w) T(cv) = cv = cT(v) T(cvw) = cvw = cT(v) T(w) T(c + v) = c+v=c+T(v) Oct(v + w) = c(v + w) = cT(v) + cT(w) T(vw) = vw = T(v) T(w)
Transcribed Image Text:Prove that the zero transformation and the identity transformation are linear transformations. (a) the zero transformation Let V and W be vector spaces, let v and w be vectors in V, let c be a scalar, and let T: V→ W be the zero transformation. Which of the following proves the zero transformation from V to W is a linear transformation? (Select all that apply.) T(vw) = 0 = 0·0= T(v) T(w) T(cv) = 0 = c0 = cT(v) T(v + w) = 0 = 0 + 0 = T(v) + T(w) T(cvw) = 0 = c0 0 = cT(v) T(w) OCT(v + w) = c(0 + 0) = cT(v) + CT(w) OT(c + v) = c + 0 = c + T(v) (b) the identity transformation Let V be a vector space, let v and w be vectors in V, let c be a scalar, and let T: V→ V be the identity transformation. Which of the following proves the identity transformation from V to V is a linear transformation? (Select all that apply.) T(v + w) = v + w = T(v) + T(w) T(cv) = cv = cT(v) T(cvw) = cvw = cT(v) T(w) T(c + v) = c+v=c+T(v) Oct(v + w) = c(v + w) = cT(v) + cT(w) T(vw) = vw = T(v) T(w)
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