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5.2 Rank and Nullity Let T : V ? W be a linear transformation and assume that V is finite dimensional. The kernel of T is a subspace of V and its dimension is called the nullity of T, i.e. nullity(T) = dim(ker(T)). The theorem on the dimension of subspaces implies that the nullity is finite and nullity(T) ? dim(V ). The dimension of the image of T is called the rank of T, i.e. rank(T) = dim(im(T)). It is not clear a priori that the rank is necessarily finite; we shall prove this below. The rank and the nullity of a linear transformation are closely related to each other Rank-Nullity Theorem Let T : V ? W be a linear transformation and assume that V is finite dimensional. Then nullity(T) + rank(T) = dim(V ). Solution 5.2 Rank and Nullity Let T : V ? W be a linear transformation and assume that V is finite dimensional. The kernel of T is a subspace of V and its dimension is called the nullity of T, i.e. nullity(T) = dim(ker(T)). The theorem on the dimension of subspaces implies that the nullity is finite and nullity(T) ? dim(V ). The dimension of the image of T is called the rank of T, i.e. rank(T) = dim(im(T)). It is not clear a priori that the rank is necessarily finite; we shall prove this below. The rank and the nullity of a linear transformation are closely related to each other Rank-Nullity Theorem Let T : V ? W be a linear transformation and assume that V is finite dimensional. Then nullity(T) + rank(T) = dim(V )..

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5.2 Rank and Nullity Let T : V ? W be a linear transformation and assume that V is finite dimensional. The kernel of T is a subspace of V and its dimension is called the nullity of T, i.e. nullity(T) = dim(ker(T)). The theorem on the dimension of subspaces implies that the nullity is finite and nullity(T) ? dim(V ). The dimension of the image of T is called the rank of T, i.e. rank(T) = dim(im(T)). It is not clear a priori that the rank is necessarily finite; we shall prove this below. The rank and the nullity of a linear transformation are closely related to each other Rank-Nullity Theorem Let T : V ? W be a linear transformation and assume that V is finite dimensional. Then nullity(T) + rank(T) = dim(V ). Solution 5.2 Rank and Nullity Let T : V ? W be a linear transformation and assume that V is finite dimensional. The kernel of T is a subspace of V and its dimension is called the nullity of T, i.e. nullity(T) = dim(ker(T)). The theorem on the dimension of subspaces implies that the nullity is finite and nullity(T) ? dim(V ). The dimension of the image of T is called the rank of T, i.e. rank(T) = dim(im(T)). It is not clear a priori that the rank is necessarily finite; we shall prove this below. The rank and the nullity of a linear transformation are closely related to each other
Rank-Nullity Theorem Let T : V ? W be a linear transformation and assume that V is finite dimensional. Then nullity(T) + rank(T) = dim(V ).

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5.2 Rank and Nullity Let T V W be a linear transformation and .pdf

  • 1. 5.2 Rank and Nullity Let T : V ? W be a linear transformation and assume that V is finite dimensional. The kernel of T is a subspace of V and its dimension is called the nullity of T, i.e. nullity(T) = dim(ker(T)). The theorem on the dimension of subspaces implies that the nullity is finite and nullity(T) ? dim(V ). The dimension of the image of T is called the rank of T, i.e. rank(T) = dim(im(T)). It is not clear a priori that the rank is necessarily finite; we shall prove this below. The rank and the nullity of a linear transformation are closely related to each other Rank-Nullity Theorem Let T : V ? W be a linear transformation and assume that V is finite dimensional. Then nullity(T) + rank(T) = dim(V ). Solution 5.2 Rank and Nullity Let T : V ? W be a linear transformation and assume that V is finite dimensional. The kernel of T is a subspace of V and its dimension is called the nullity of T, i.e. nullity(T) = dim(ker(T)). The theorem on the dimension of subspaces implies that the nullity is finite and nullity(T) ? dim(V ). The dimension of the image of T is called the rank of T, i.e. rank(T) = dim(im(T)). It is not clear a priori that the rank is necessarily finite; we shall prove this below. The rank and the nullity of a linear transformation are closely related to each other
  • 2. Rank-Nullity Theorem Let T : V ? W be a linear transformation and assume that V is finite dimensional. Then nullity(T) + rank(T) = dim(V ).