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# a - The Chapter 4 Super Summary Use this as a quick...

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The Chapter 4 Super Summary Use this as a quick reference guide if you’re in a hurry. But to really understand the material, or for helpful examples (and non-examples), please read the chapter summaries below it! Formatting note : writing math in a google document is kind of hard to deal with sometimes. When I write something like , I’m calling a three-row vector with vertical entries 0,1,2. Chapter 4.1. A set is a vector space if, for any and in , and any scalar , is in , and is in The span of vectors is the space of all linear combinations of these vectors, i.e. the vectors of the form: where are scalars. Chapter 4.2. A linear transformation is a function from one vector space to another such that, for any and any scalar , Every linear transformation from to corresponds to an matrix, and vice versa. A linear transformation has four key related spaces: Its domain Its codomain Its range , the subspace of that’s the collection of all possible outputs of Its kernel , the subspace of that’s the collection of all inputs that maps to the zero vector A linear transformation is one-to-one if its kernel is just the zero vector. It’s onto if its range is the same as its codomain. It’s invertible if it’s both. The column space of a matrix is the span of its column vectors, and the null space of is the space of all vectors such that , i.e. the space of all solutions to the homogeneous system.

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If is a linear transformation and is its corresponding vector, then the range of is the column space of and the kernel of is the null space of . Chapter 4.3. A collection of vectors is linearly dependent if there exist scalars , that aren’t all zero such that: A collection that’s not linearly dependent is called linearly independent. For any matrix , the columns of form a linearly independent collection of vectors if and only if the only solution to is the trivial . A basis of a vector space is a collection of vectors such that: is linearly independent Be sure to check my summary of this chapter for instructions on how to find bases for the null space, column space and row space! Chapter 4.4. Say is a basis of a vector space . For any , there is a unique -coordinate form , where if then: If is a basis of the space , then the change-of-coordinates matrix is the matrix with columns . This matrix is useful because: is always invertible. Chapter 4.5. Every basis of a vector space has the same number of vectors, and that number is called its dimension . If you have a set of vectors in , you can get some information them based on : If , then is linearly dependent. It may or may not span . If , then either is a basis of , or it’s neither linearly independent nor does it span .
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a - The Chapter 4 Super Summary Use this as a quick...

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