Essentially linear algebra, the source video bilibili
I had always felt did not really understand linear algebra, linear algebra for learning basically rely on memory rather than understanding, in order to seriously study linear algebra, linear algebra behind clarify the nature, hereby learn to do down notes.
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What is a vector
Linear algebra is the most basic and most part of the root cause is the vector, so the vector is what we need to reach a consensus and move on.
Vector, usually for understanding vector for different people in different areas, because in each of these areas in the form of application and differ, in general, there are three different understanding:
- In Physics opinion, is the vector space with direction arrows, determined a vector and its length is referred to
- In computer science view, the vector is an ordered list of numbers, for example, in the analysis we look at housing prices and housing area price, then there will be (housing area, price) of such a vector
- In mathematics it seems, the vector can be anything, as long as two vectors and vector sum of numbers can be multiplied meaningful
In this process, the vector and the vector is multiplied by a digital addition and these two operations is important.
We consider a vector, is a vector
from the origin of the arrow of departure, we can separately from the physical point of view and perspective on this computer vector.
The coordinates of a vector formed by a pair of numbers, each number corresponds to a single vector, each vector is represented by a unique pair of numbers.
Definition of vector addition:
we can seen as a motion vector, i.e., do some movement toward a certain direction, and the adder corresponds to a motion vector on the superposition of two vectors, as shown below.
From the perspective figures, vector addition is adding the number corresponding to the position.
Vector multiplication:
in digital and vector multiply in the process, there is no digital direction is called a scalar (Scalars), the main role is played by digital zoom vector.
Linear transformation, the space spanned by the base
Based strictly defined:
A set of basis vectors span space is a space independent of the linear vector set.
Linear transformation
We are all familiar with the coordinates, we are now in a new perspective on the coordinates: We each coordinate as a scalar, they will be scaling a vector.
In the plane, there are two special vector which points to the right (positive x-axis direction or points) are the unit vector i ^ units, and just above the point J ^ .
At this point we can put （3，-2）= 3 * i + （-2） * j viewed as two after scaling the vector sum.
Remember, Scalable Vector and adding this concept.
In fact, at this time we have iand jvector called basis vectors xy coordinate system.
If we choose a different basis vectors what will happen?
The answer is selected as the base vectors of any two vectors (not collinear), we can obtain all the vectors on a plane.
When we describe a vector number, which we rely on basis vectors currently used.
And the two vectors is referred to these two vectors a linear combination .
Why is it called a linear combination of it? We provide an idea:
- When we let the sum of two vectors, if a fixed vector, another vector so that any movement of the end of the generated vectors describe a straight line.
And if two scalar same time change, we will be able to get all the vectors!
Vector space
Definition: can be represented as a set orientation to the vector quantity of a linear combination, it referred to a given vector space spanned by (span).
Any two-dimensional non-collinear vectors, their space spanned the entire two-dimensional plane;
and for the two-dimensional vector collinear, their space spanned is a straight line.
Actually, the problem is drawn vector space, simply by multiplying the number of vector addition and vector two operations, what is the set of all vectors you may get yes.
Vector point :
When we consider many vectors, usually when we use the end of the vector represents a vector, because the starting point is the origin, when we consider all the two-dimensional vector, we need only consider the infinite two-dimensional plane can be.
When we consider a few vectors, we can still consider the vector into a vector with an arrow.
Three-dimensional vector space Zhang Cheng
When we have to consider three-dimensional space when Zhang Cheng vector, the problem becomes interesting.
If we fix a vector which does not move, the other two vectors to move freely, adding three vectors, we can get a plane.
Three linear combination of vectors is to choose three scalar for scaling the three vectors, respectively, and the results are added, to obtain a linear combination of three vectors.
A linear combination of all three vectors formed their sheets into space.
When we time plus the first three vectors, two vectors before the plane formed by the third movement along the direction vector in space, until it covers all the positions in the space (corresponding to a surface moving along a direction , will eventually occupy all positions).
A plurality of linear correlation vectors
Combine said before, we have two more perspective to understand linear correlation vectors:
- If there are multiple vectors, if you remove one of them without affecting the vector space spanned by them, then we call this vector and the vector before the linear correlation
- If there is a vector that can be expressed as a linear combination of other vectors, then call this vector and vector linear correlation before, because this vector has lagged behind other vector spanned space.
On the other hand, if each vector has contributed both to span the space, then they say they are linearly independent.
After learning matrix will continue its operations and related properties.