Questions and Answers
Let T: V -> W be a linear transformation with kernel K. The dimension of the kernel of T is called the _______ of T.
nullity
Let T: V -> W be a linear transformation with range K. The dimension of the Range of T is called the ____ of T.
rank
What does the Dimension Theorem state?
The nullity of T plus the rank of T is equal to the dimension of V.
Suppose T: V -> W is a linear transformation, and suppose dim(V) = dim(W). Then T is one-one if and only if (IFF) T is onto, and conversely ______ (_) = __.
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Recall f: A -> V is - to mean that different elements in domain A give different functions f(x).
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Suppose T(v1) = T(v2), v1 = v2 since T is one-one. Let x be in the Kernel (T), then T(x) = 0.
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Suppose Kernel (T) = 0, prove T is -.
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Suppose V and W have the same dimension. Then, T is ____ if and only if (IFF) Range (T) = W.
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T: V -> V is called a ______ ________ (on V).
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Suppose T: V -> V is a linear operator on V. We define T _______ as follows: y = T^-1(x) IFF T(y) = x.
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Study Notes
Nullity
- Nullity of a linear transformation T: V -> W is the dimension of the kernel (null space) of T.
- The kernel is the set of all vectors in the domain V that map to the zero vector in the codomain W.
Rank
- Rank of a linear transformation T: V -> W is the dimension of the range of T (the image of T).
- The range consists of all possible outputs of the transformation from V to W.
Dimension Theorem
- For a linear transformation T: V -> W, the relationship between Nullity and Rank is given by: dim(Kernel) + dim(Rank) = dim(V).
- This theorem connects the dimensions of the kernel, image, and the whole domain.
Kernel and One-One Theorem
- If dim(V) = dim(W) for a linear transformation T: V -> W, then T is one-to-one (injective) if and only if it is onto (surjective).
- Applies to general functions, indicating a direct correlation between injectivity and surjectivity under equal dimensions.
One-One (Injective) Definition
- A function f: A -> V is one-one if different elements in the domain A yield different outputs in V.
- Mathematically, if f(x) = f(x'), then x must equal x'.
Kernel Proof for One-One
- For T: V -> W, if T(v1) = T(v2), then v1 must equal v2, proving T is one-one.
- If x is in the kernel, T(x) = 0 and since T is one-one, x must be the zero vector.
One-One Converse
- If the kernel of T is zero, then T is one-to-one.
- Proof involves showing that if T(v1) = T(v2) leads to v1 = v2, thereby establishing injectivity.
Onto Converse
- If dimensions of V and W are equal, T is onto if and only if its range is equal to W.
- Using the dimension theorem, if the range of T covers W, then the kernel must be zero, indicating T is also one-one.
Linear Operator Definition
- A linear transformation T: V -> V is called a linear operator.
- Linear operators map a vector space to itself while maintaining linearity.
Inverse Definition
- For a linear operator T: V -> V, the inverse T^-1 is defined such that y = T^-1(x) if and only if T(y) = x.
- Under the condition that T is both one-to-one and onto, T^-1 exists, indicating a bijective correspondence within the same dimensional space.
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This quiz provides essential flashcards focusing on the concepts of nullity and rank within linear transformations. Explore definitions and understand the significance of these terms in linear algebra to strengthen your comprehension of the subject.
