Intuitive explanation of the Fundamental Theorem of Linear Algebra

Question

Can someone explain intuitively what the Fundamental Theorem of Linear Algebra states? and why specifically it is called the above? Specifically, what makes it 'Fundamental' in the broad scope of the theory.

Answer 1

Imagine a projection, for example, from the whole $\mathbb{R}^3$ to the $x$-$y$ plane. It compresses each line that parallels to $z$-axis to a point on the plane. So there is a one-to-one relationship between the lines and the points. Notice that the lines are the translations of the $z$-axis -- that is also what quotient means. And the $z$-axis, is just the kernel of the projection, so we can see that $\operatorname{im} A\simeq V/\ker A$.

As for the dimension, the dimension of $\ker A$ measures how much we compress, while the dimension of $\operatorname{im} A$ measures how much we leave -- the amount we compress, plus the amount we leave, equals the whole thing, intuitively.