Let V and W be real vector spaces and T be a linear transformation from V into W, that is, T(au + be) = aT(u) + bT(v) for vector u and v in V and scalars a and b. The set of vectors in V so that T(v) = 0 is called the null space of T, which is a subspace of V, and the set of T(V), which is a subspace of W, is called the range of T. Similar to the matrix case, we have dim (null(T))+dim range (T) = dim (V). T(0)=?. If T(v) = w, then what is T(3v)? If T(u) = w and T(v) = x, find T(2u - 3v).
Let V and W be real vector spaces and T be a linear transformation from V into W, that is, T(au + be) = aT(u) + bT(v) for vector u and v in V and scalars a and b. The set of vectors in V so that T(v) = 0 is called the null space of T, which is a subspace of V, and the set of T(V), which is a subspace of W, is called the range of T. Similar to the matrix case, we have dim (null(T))+dim range (T) = dim (V). T(0)=?. If T(v) = w, then what is T(3v)? If T(u) = w and T(v) = x, find T(2u - 3v).
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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Transcribed Image Text:Let V and W be real vector spaces and T be a linear transformation from V into W, that is, T(au + bv) = aT(u) +bT(v)
for vector u and v in V and scalars a and b. The set of vectors in V so that T(v) = 0 is called the null space of T, which is
a subspace of V, and the set of T(V), which is a subspace of W, is called the range of T. Similar to the matrix case, we have
dim (null(T)) + dim range(T) = dim (V).
T(0) =?. If T(v) = w, then what is T(3v)? If T(u) = w and T(v) = x, find T(2u - 3v).
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