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 a column vector space, and Also the 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

Note that in the first example of a linear combination, there are only two columns in a (in this case the Gaussian elimination method must be singular, the number of pivots is not enough)

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 linear combination of coefficients, a little difference

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.

   The relationship between the column vector space and the nullspace, and the relationship between the complete solution and the complete, see the section, when the equation is solved the basic transformation changes the column vector space, but the solution is not changed.

   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

2. Solution Ax=0 and Ax=b

1. For invertible matrices, the solution of ax=0 is only x=0,ax=b for any B. When there is a non-0 solution in nullspace or a column vector space is small, then we sectionto the situation of processing.

   Echelon form (Echelon form):

   Because we use the elimination method to get U, we will restore it in the reverse process, naturally we can get the original matrix A, so:

   L is a phalanx, the number of rows equals the number of rows A and U

   Pa=lu is the general form:

   Get R: 1. Turn the pivots of U into 0 2. From the bottom to the pivots directly above the element into 0, as follows:

   * Matlab gets R r=rref (A), for invertible matrices, r=i

   The above chapter is the Pa=ldu,gauss-jordan method [a i]->[u l inverse]->[i a inverse], involving the a->u,pivots above the element 0, and so on, the method of R and the two methods of likeness.

   R still contains an identity matrix.in the pivot rows and pivot columns.

   Rx=0 ux=0 ax=0 have the same solution.

   Pivot variables: variables (in vector x) correspond to columns with pivots

   Free variables: variables (in vector x) correspond to columns without pivots

   As can be seen below, the pivot varialbes in the representation of X is represented by free variables. And the free variables is called the variables because these variables can be arbitrarily evaluated.

   Using R for Ax=0:

   Special solutions for ax=0: Each column on the right side of the ③ is a special solutions.

   A's nullspace can be seen as a collection of combination of special solutions.

   In the above example, the nullspace of a is a vector space of 2 dimensions.

   Note: Careful observation, the number of special solutions equals the number of free variables

   For a (m by n,n>m), there is at least a n-m free variables.

   A very important conclusion:

   From the process of rx=0, we can see that each extra free variables, there will be an extra special solution, for a (m by N,n>m), at least n-m a free variables, which is the reason for the 2C. And because free variables can be arbitrary value, so the number of solutions is infinite.

   Solving Ax = b, U x = c, and Rx = d

   and ax=0 different point: even if n>m (more unkonwns than equations), there is not necessarily a solution. The b!=0 on the right side of the equation causes the equation to be parallel (b=0 the equation coincides).

   To explain the order of the problem: first to find a solution to the equation B, the main use of B should be in the column vector space method. Use the same method as in the previous section to find the solution of the equation (using free variables and pivot variables) to draw a conclusion.

2. Ask Ax=b:

   Conclusion:

   Every solution to Ax = B are the sum of one particular solution and a solution to Ax = 0:

   Regarding, B can be regarded as 0+b, according to the solution of the b=0 part of the previous section method, the solution of B=b part is. (from the phenomenon to the conclusion, this is a generalization)

   There is also a special solution when b=0, except that the special solution is 0.

   After ax=b into Rx=d, it is possible to draw conclusions directly, paying attention to this method, along with the method of directly obtaining results from the previous section:

   The d median corresponds to the special solution for the position of the pivot variable, where the free variable corresponds to a 0.free variable corresponding to the special solution.

   Rank of the matrix: the rank of matrices

   You see how the rank r is crucial. It counts the pivot rows in the "row space" and the

   Pivot columns in the column space. There is n−r special solutions in the Nullspace.

   There is m−r solvability conditions on B or C or D (the reason for this is that the remaining M-r rows determine whether the equation has a solution).

   Problem set:

   1.

   The equation can still have nullspace when there is no solution, there is xn but no XP at this time.

   2.

   Ans

   3.

   That is, the column where pivot columns is a linearly independent column

   4.[??]

   Less than a good proof (AB's result is a linear combination, can not escape a where the ' plane '), less than B?

   5. Note If the ab=i, in the AB is not a phalanx can also be set up, the definition of left inverse right inverse of the formula are all refers to the square, not the case of the square is not equal to the right inverse (the dimension is not the same), the definition of inverse is at the same time to meet the left inverse right inverse is called inverse.

   6.

   7.[!!]

   8. A system ax=b have at the most one particular solution: wrong. Its arbitrary solution can be recognized as a special solution. As long as Xp+xn=x set up can be.

   9. This is probably the case.

   (a) Rows>rank (a), Columns=rank (a) (b). Rows=rank (a), Columns>rows (c) Rows>rank (a), Columns>rank (a) (d) Invertible

   10.

   It is easy to use the above conclusion: (a) rank 2 and the solution is (b) rank is less than the number of columns, no solution or infinite solution.

   11.

   The equations intersect into points, there is no unknown quantity in the equation group, intersect into a line, there is an unknown amount, so the rank is 3.

   (b)

   12. Pivot variable in the rear column is not available for use in the Forefront free variable.

   Whydoes no. 3 by 3 matrix has a nullspace that equals its column space

   Reversible: Nullspace has only one element, column space is

   Rank 2 o'clock: Nullspace is a line, column space is a polygon

   Rank 1 o'clock: Nullspace is a polygon, column space is a line

Chapter II Vector space