Linear Algebra - Column Space, Zero Space

The column space is concerned with the set b that makes Ax = b, and the null space is concerned with the value of x when b is a zero vector

Column space
A is the following matrix
????? 123411112345 ?????
Because A is a three-dimensional vector, it is impossible to cover all four-dimensional space, so for the Ax = b this formula, there will be b does not exist, we are looking for when b exists.

According to the general thinking, first study b is a zero vector, this time Ax = b is definitely established. In addition there is no situation? Certainly there is, when b is a in a column or a column c times, it is also established. That is to say, when b matrix is ​​a linear combination of a column of A matrix, Ax = b holds.

Zero space Zero space is concerned: when b is zero vector x value. It is easy to know that Ax = b holds true for x when it is a zero vector, since any vector multiplied by a zero vector results in a zero vector. Then there is no established situation? Note that the above A matrix, which adds up the first two columns is equal to the third column, so (column one plus two minus three) is also a zero vector. The same c * (column one plus two minus three) is also established. That is to say at this time makes Ax = b established x =. Zero space at this time for an origin in R3 straight line.