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.) O T(cvw) = 0 = c0 · 0 = cT(v) T(w) O T(c + v) = c + 0 = c + T(v) T(yw) = 0 = 0.0 = T(v) T(w) O T(v + w) = 0 = 0 + 0 = T(v) + T(w) O T(cv) = 0 = c0 = cT(v) a CT(v + w) = c(0 + 0) = cT(v) + CT(w) (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.) O T(cv) = cv = cT(v) O T(cvw) = cvw = cT(v) T(w) a CT(v + w) = c(v + w) = cT(v) + CT(w) O T(v + w) = v + w = T(v) + T(w) O T(vw) = vw = T(v) T(w) O T(C + v) = c + v = c + T(v)
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.) O T(cvw) = 0 = c0 · 0 = cT(v) T(w) O T(c + v) = c + 0 = c + T(v) T(yw) = 0 = 0.0 = T(v) T(w) O T(v + w) = 0 = 0 + 0 = T(v) + T(w) O T(cv) = 0 = c0 = cT(v) a CT(v + w) = c(0 + 0) = cT(v) + CT(w) (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.) O T(cv) = cv = cT(v) O T(cvw) = cvw = cT(v) T(w) a CT(v + w) = c(v + w) = cT(v) + CT(w) O T(v + w) = v + w = T(v) + T(w) O T(vw) = vw = T(v) T(w) O T(C + v) = c + v = c + T(v)
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.) O T(cvw) = 0 = c0 · 0 = cT(v) T(w) O T(c + v) = c + 0 = c + T(v) T(yw) = 0 = 0.0 = T(v) T(w) O T(v + w) = 0 = 0 + 0 = T(v) + T(w) O T(cv) = 0 = c0 = cT(v) a CT(v + w) = c(0 + 0) = cT(v) + CT(w) (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.) O T(cv) = cv = cT(v) O T(cvw) = cvw = cT(v) T(w) a CT(v + w) = c(v + w) = cT(v) + CT(w) O T(v + w) = v + w = T(v) + T(w) O T(vw) = vw = T(v) T(w) O T(C + v) = c + v = c + T(v)
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.)
V T(cvw) = 0 = co · 0 = cT(v) T(w)
V T(C + v) = c + 0 = c + T(v)
VT(vw) = 0 = 0· 0 = T(v) T(w)
V T(v + w) = 0 = 0 + 0 = T(v) + T(w)
V T(cv) = 0 = c0 = cT(v)
M cT(v + w) = c(0 + 0) = cT(v) +
cT(w)
(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.)
V T(cv) = cv = cT(v)
V T(cvw) = cvw = cT(v) T(w)
M cT(v + w) = c(v + w) = cT(v) +
CT(w)
V T(v + w) = v + w = T(v) + T(w)
V T(vw) = vw = T(v) T(w)
V T(c + v) = c + v = c + T(v)
Branch of mathematics concerned with mathematical structures that are closed under operations like addition and scalar multiplication. It is the study of linear combinations, vector spaces, lines and planes, and some mappings that are used to perform linear transformations. Linear algebra also includes vectors, matrices, and linear functions. It has many applications from mathematical physics to modern algebra and coding theory.
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