Chapter II Vector space

1. Vector spaces and sub-spaces

: Contains all n-dimensional column vectors (the reason for using R: The elements of a column vector are real numbers)

Operations that are supported within all vector spaces:

Real vector space:

8 Rules:

The use of more or the definition of vector space and subspace, that is, the result of multiplication and addition is still within the collection. SubSpace is also a vector space

Conversely, if a set cannot form a vector space, the result of adding or multiplying the elements of the collection may not be within the set.

Sub-space:

The 0 Vector is a subspace of any vector space.

The smallest subspace consists of only one vector: 0 vectors. Called a 0-dimensional space. Vector space is not allowed to be an empty set.

The largest subspace is the original vector space itself.

SubSpace of 3-dimensional space:

The subspace can shrink the dimension (by), but the essence is a subset drawn from the original set of spatial points and the elements in the collection come from the original collection.

Column vector space:

Note that the atom of the vector space is ' column ' (the conceptual column, the column can actually be in various forms, such as a function, a matrix).

The relation between the equation solution and the column vector space:

For

Ax = B can be solved if and just if B lies in the plane it is spanned (calibrated) by the and the vectors. This plane is the column vector space, but also Yes subspace (over 0 points). This plane is the proof of subspace P89

C (a): The column vector space of a, is the subspace that proves p89. Note that we are discussing the complete works of the various problems, similar to the relationship with 1 belonging to the natural number N.

N (a): A of the nullspace, is the subspace, which proves:

X+x ' within Nullspace, CX is also within Nullspace, so the complete universe of X is subspace

Notice at the beginning of the example of a linear combination, A is only two columns (in this case, the Gaussian elimination method must be a singular case, the number of pivots is not enough!) ), when the column vector space is a polygon, a can also have three columns of-----

Nullspace:

Note: the previously said vector space is directly composed of B, nullspace does not seem to be so, it is composed of 0 vector space of linear combination of coefficients, a little difference, and if a is less than the number of rows, it seems to see B more intuitive.

Independent, or linearly irrelevant.

Understand the purpose of C (a) and N (a):

Problem set:

1. [??] Give one or several elements to the minimum vector space containing these elements problem set2.1 14
2. The subspace consists of four types: itself, polygon, line, 0.

3. [??] B's Column vector space has been different from a, what is the relationship between column vector space and reconciliation? The column vector space for C is still the same as a.

   4.A can be uniquely determined by two vectors, but cannot be determined by three vectors alone

   .

   C and D can determine any number of vectors such as E and E ', E in and B to determine a.

   Unless b is in the vector space of a, the second figure above shows that the addition of a d in a vector space is still solvable.

4. The invertible equivalent to the non-singular, the non-singular equivalent to the matrix column vectors are independent of each other (linearly independent), then the linear space of the invertible matrix is

Chapter II Vector space