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3. Let V and W be F-vector spaces, and let L(V, W) be the set of all linear transfor- mations from V to W. For any S, TE L(V, W) and c E F, we define S+T and cT as in Problem 2. Show that L(V, W) is an F-vector space. What is the zero vector in L(V, W)?

3. Let V and W be F-vector spaces, and let L(V, W) be the set of all linear transfor- mations from V to W. For any S, TE L(V, W) and c E F, we define S+T and cT as in Problem 2. Show that L(V, W) is an F-vector space. What is the zero vector in L(V, W)?

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.4: Linear Transformations
Problem 24EQ
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2. Let V and W be vector spaces over a field F. If S and T are any mappings from V to W, and c E F is any scalar, we define S+T: V → W and cT: V → W by (S+T)(v) = S(v) + T(v) and (cT) (v) = c · T(v) for all v € V. (a) Show that if S and T are both linear, then S + T is linear. (b) Show that if T is linear, then cT is linear.
Transcribed Image Text:2. Let V and W be vector spaces over a field F. If S and T are any mappings from V to W, and c E F is any scalar, we define S+T: V → W and cT: V → W by (S+T)(v) = S(v) + T(v) and (cT) (v) = c · T(v) for all v € V. (a) Show that if S and T are both linear, then S + T is linear. (b) Show that if T is linear, then cT is linear.
LB. 3. Let V and W be F-vector spaces, and let L(V, W) be the set of all linear transfor- mations from V to W. For any S, T = L(V, W) and c E F, we define S+T and cT as in Problem 2. Show that L(V, W) is an F-vector space. What is the zero vector in L(V, W)?
Transcribed Image Text:LB. 3. Let V and W be F-vector spaces, and let L(V, W) be the set of all linear transfor- mations from V to W. For any S, T = L(V, W) and c E F, we define S+T and cT as in Problem 2. Show that L(V, W) is an F-vector space. What is the zero vector in L(V, W)?
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