2-D is a plane, which is an area composed of length and width (or partial curves). Three dimensions are the volume formed by two dimensions plus height. Four dimensions are divided into four dimensions in time and space. People often refer to the transfer of objects on the timeline.
The dimension of the solution space of homogeneous linear equations is the number of vectors contained in the basic solution system; That is, n-r(A).
The main problems discussed in linear equations are:
① When will the equations be solved?
(2) The number of solutions to the equation.
③ Solve the equations with solutions and determine the structure of the solutions.
These problems have been satisfactorily solved: if the given equations have solutions, then rank (A)= rank (augmented matrix); If rank (A)= rank = r, then there is a unique solution when r=n; When r < n, there are infinite solutions; It can be solved by elimination.
Extended data:
Let W be a nonempty set of linear space V over field P. If addition in V and scalar multiplication of fields P and V form a linear space over field P, W is called a linear subspace (or vector quantum space) of V, or simply a subspace.
Note: The necessary and sufficient condition that the nonempty subset W of V is a subspace is:
The sum α+β of any two vectors α and β of (1) subset W is still a vector in W;
(2) The product kα of any number k of field P and any vector α of subset W is still a vector in W. ..
In linear space, a subset composed of a single zero vector is a linear subspace, which is called zero subspace.
The linear space V itself and a single zero vector are both linear subspaces of V, and these two special subspaces are called ordinary subspaces of V; Linear subspaces other than trivial subspaces are called nontrivial subspaces of V.
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