Solution#
Use what you have learned from previous sections to answer the following questions:
What is the difference between injective, surjective and bijective?
Answer
Injective (one-to-one) means that an element in \(Y\) cannot be a mapping of more than 1 element from \(X\). We can write it as:
\[f(x_1) = f(x_2) \leftrightarrow x_1 = x_2\]Surjective (onto) means that all elements in \(Y\) is a mapping of an element in \(X\) through \(f\). This is the case when range of \(f\) spans the entire co-domain. We write this as:
\[\forall y \in Y, \exists x \in X \ni y = f(x)\]Bijective is both injective and surjective, i.e. range spans entire co-domain, and for every \(y\) there is a unique \(x\) that maps to \(y\) through \(f\).
When will the range of a function equals the co-domain?
Answer
When the function is surjective or bijective.
Given \(\mathcal{f}: X \rightarrow Y\), when will \(\mathcal{f}\) be called a function?
Answer
\(f\) is a function when the mapping from \(X\) to \(Y\) is unique, i.e. there exist a unique value of \(f(x) \forall x \in X\).
What is a field?
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What are the rules of each of the operation of a field?
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What is a vector space?
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What are the rules of a vector space? What is the difference between these rules between the vector space and a field?
Hint
Consider the multiplication operation, and what type does it operate on for a field, and for a vector space
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How to prove a space is a vector space?
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What is a subspace?
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Definition of linear independence?
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What is a basis?
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What is coordinate?
What is the relationship between coordinate and basis?
Are coordinates unique?
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How many basis can a vector space have? What is the dimension of a vector space?
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What is the definition of a linear map? What does superposition mean?
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What is range space, null space? What is another name for range space and null space?
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Which vector space does range space belong to? Which vector space does null space belong to?
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Given a linear equation \(\mathcal{A}(u) = b, with b \in V\), when will \(b \in \mathcal{R}(\mathcal{A})\)?
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Given a linear equation \(\mathcal{A}(u) = b, with b \in \mathcal{R}(\mathcal{A})\), when will there be a unique solution?
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