I’m just getting started with linear algebra and I’m not sure what column spaces or null spaces actually signify. What is the geometrical interpretation of both?
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Do you know how to multiply a matrix by a vector?
For an m x n matrix, its column space is *all* the linear combinations of the columns. What that means in terms of matrix-vector multiplication is that the column space is *all* the vectors A**v** as **v** runs over **R**^(n). In other words, if you use the matrix A to define a function from **R**^(n) to **R**^(m) that sends each **v** to A**v**, then the column space is the range of that function.
The null space is all the **v** where A**v** is the zero vector. It tells you how much A “collapses” **R**^(n) when you apply it to vectors there: A**u** = A**w** exactly when A(**u** – **w**) is the zero vector, meaning **u** – **w** is in the null space of A.
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