Core, range, vector space, row space, 0 space

1. Nuclear

all the transformed matrices become a set of vectors consisting of 0 vectors , usually represented by Ker (A).

Suppose you are a vector, there is a matrix to transform you, if you unfortunately fall into the nucleus of this matrix, then it is regrettable that after the conversion you become a void of 0. In particular, it is noted that the concept of "transformation" (Transform) is verified, and a similar concept in matrix transformation is called "0 space". Some materials use t to refer to the transformation, when the matrix is connected with a, the matrix is directly regarded as "transformation". The space in which the nucleus is located is defined as v space, which is the original space of all vectors.

2. Domain value

A set of vectors formed by the transformation matrix of all vectors in a space, usually represented by R (A).

Suppose you are a vector, there is a matrix to transform you, and the range of this matrix represents all the possible positions for you in the future. The dimension of the range is also called rank. The space where the range is located is defined as W space.

3. Space

The vector and the multiplication and multiplication operations on it make up the space. Vectors can be transformed (and only in) space. Use the coordinate system (base) to describe the vector in space.

Both nuclear and domain, they are closed. This means that if you and your friend are trapped inside the nucleus, you will not be in the nucleus, either by adding or multiplying, and this will constitute a subspace. The same range of domains.

The mathematician proves that the dimension of V (the space defined by the nucleus as V space) must be equal to the dimension of the kernel of any of its transformation matrices plus the dimension of the domain.

Strict proof can refer to the relevant information, here is an intuitive proof method:

The dimension of V is the number of bases of V. These bases are divided into two parts, one part in the nucleus and the other as the primary image of the non-0 image of the range (which can certainly be divided, because both the nucleus and the range are separate subspace). If any vector in V is written in the form of a base, then the vector must also be part of the kernel, and the other part in the original image of the non-0 image in the range. Now the transformation of this vector, the part of the kernel is of course zero, and the other part of the dimension is just equal to the dimension of the domain.

four, the transformation matrix row space and 0 space relations

Depending on the nature of the matrix, the number of rows of the transformation matrix equals the dimensions of V, and the rank of the transformation matrix equals the dimension of the range r, so it can be concluded that:

Because the rank of a is another dimension of the A-line space (note that this number is definitely less than the number of rows in the non-full matrix), the above formula can be changed to:

This form is written because we can find that the 0 space of a and the line space of a are orthogonal and complementary. Orthogonal is because 0 space is the kernel, and the line vector by definition multiplied by a is of course zero. The complementarity is because they add up just spanned the entire v space.

This orthogonal complementarity leads to very good properties, as the 0 space of a and the base of the line space of a can be combined to form a base of V.

the relationship between the transformation matrix column space and the left 0 space

If you transpose the above equation, you can get:

Because the actual meaning is to reverse the range and definition domain, so the 0 space is the value of the field outside the value of 0 points in the space of all vectors, it is referred to as "left 0 space" (Null spaces). This will give you:

Similarly, the left 0 space of a and the column space of A are also orthogonal complementary, they add up just can be a w space, their base also formed the base of W.

relationship between row space and column space of transformation matrices

The transformation matrix actually transforms the target vector from the row space to the column space.

The line space, column space, 0 space, and left 0 space of the matrix make up all the space in the study of linear algebra, and make clear the relationship between them, which is very important for the separate base transformation.

The secret of the characteristic equation

We are trying to construct a transformation matrix A: It transforms the vector into a range space, and the base of the range space is orthogonal; not only that, it is also required to have the form of a base V, which is a known base of the original space. This allows us to transform complex vector problems into an exceptionally simple space.

If the number is not equal to V, then replace A, can become a symmetric and semi-positive matrix, its eigenvector is the required base v!

Again, the matrix is not equal to the transformation, and the matrix as a transformation simply provides a way to understand the transformation matrix. Alternatively, we can assume that the matrix is just a form of transformation.

Turn from: geometric and physical meanings of eigenvalues and eigenvectors

Core, range, vector space, row space, 0 space