2. Linear maps Consider the following three vectors in R³: --0) --(9) +-0. V1 V3 = (a) Show that = (V₁, V₂, V3) is a basis for R³.

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Chapter2: Second-order Linear Odes
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2. Linear maps
Consider the following three vectors in R³:
~-0)- --(9)· --0)
(6).
V1
=
V3 =
(a) Show that = (V₁, V2, V3) is a basis for R³.
Consider the following four vectors in R¹:
--0-0-0-0
(b) Show that = (U₁, U2, U3, U4) is a basis for R¹.
A linear map T: R4R³ is determined by:
T(u₁)= V₁, T(u₂) = 2v2, T(u3) = T(U4) = 0,
(c) Give the standard matrix of T, i.e., in terms of the standard bases for R4 and R³.
(d) Give a basis for the image of T.
(e) Give a basis for the kernel of T.
Transcribed Image Text:2. Linear maps Consider the following three vectors in R³: ~-0)- --(9)· --0) (6). V1 = V3 = (a) Show that = (V₁, V2, V3) is a basis for R³. Consider the following four vectors in R¹: --0-0-0-0 (b) Show that = (U₁, U2, U3, U4) is a basis for R¹. A linear map T: R4R³ is determined by: T(u₁)= V₁, T(u₂) = 2v2, T(u3) = T(U4) = 0, (c) Give the standard matrix of T, i.e., in terms of the standard bases for R4 and R³. (d) Give a basis for the image of T. (e) Give a basis for the kernel of T.
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