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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).

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.2: Linear Independence, Basis, And Dimension
Problem 43EQ
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Kishan
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).
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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