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Let V and W be vector spaces over a field F. If S and T are any mappings from V to W, and c € 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.

Let V and W be vector spaces over a field F. If S and T are any mappings from V to W, and c € 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.

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.
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