**v**and

**w**, a vector space is formed by the set of all linear combinations formed between

**v**and

**w**, namely

*c*v +

*d*

**w**for arbitrary coefficients

*c*and

*d.*

*Columns spaces and null spaces are special categories of vector spaces that have interesting properties related to systems of linear equations,*

*A*

**x**=

**b**. The column space of the matrix A, C(

*A*), is simply the linear combinations of

*A*'s column vectors. This implies that

*A*

**x**=

**b**may only be solved when

**b**is a vector in A's column space. Finally, the null space of matrix

*A*is another vector space formed by all the solutions to

*A*

**x**=

**0**.

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