First, the vector space
Linear algebra is the study of a vector and matrix math, matrix vector is constructed so linear algebra is to study a science vectors, vector spaces and vector linear combination of properties.
We know that there are several basic vector operations, vector addition, is a vector where each component corresponds to the sum, multiplied by a scalar vector is a vector where each component of the scalar multiplication, namely:
$\mathbf{u}+\mathbf{v}=\left(u_{1}+v_{1}, u_{2}+v_{2}, \dots u_{n}+v_{n}\right)$
$k \mathbf{u}=\left(k u_{1}, k u_{2}, \ldots, k u_{n}\right)$
Let's talk about what is space, so what the basic features some space. Three-dimensional space, for example:
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- 1: a lot (actually an infinite number) location of points
- 2: There is a relationship between these opposing points
- 3: you can define the length, the angle in space
- 4: This space can accommodate movement, we are talking about here is the movement from one point to another of movement (conversion), rather than "continuous" nature of the movement in the sense of calculus
Article IV wherein: receiving movement is an essential feature space.
If as a vector space of a point, then the transformation vector is the point of this motion in space. So, the vector space is a set, this set of vector addition of several ride is closed
In other words, as long as the vector in this space, then multiply the vector in accordance with the addition of several ways movement, it would have been in this space . Therefore, the addition of several multiplication vector space enclosed space is also referred to as linear
After defining a vector space, we look at the most common type of vector space:
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- $ \ Mathbf {R} ^ {n} $ defines a dimension $ n-$ real vector space, that all dimensions of the set of real vectors $ n-$ when $ n = 2 $ is flat, when $ n = 3 $ is the familiar three-dimensional space
- It is easy to see that, if there are any two vectors $ \ mathbf {u}, \ mathbf {v} $ in the vector space, then $ \ mathbf {u} + \ mathbf {v} $ necessarily also in this space and $ c \ mathbf {u} $ is also in this space
Second, the subspace
The above describes the vector space, we look to quantum space. We know, $ \ mathbf {R} ^ {n} $ containing all dimension $ n $ is a real vector, but sometimes, we may not need to consider all of the $ n $ dimensional real vector, we need only consider the part the $ n $ dimensional real vector
Then this part of the set of real vector or spatial configuration, called subspace. Subspaces can be seen as a subset of the vector space, but is itself closed subset:
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- That is, if there are any two vectors $ \ mathbf {u}, \ mathbf {v} $ in the sub-space, then $ \ mathbf {u} + \ mathbf {v} $ is also inevitable in this subspace, and $ c \ mathbf {u} $ also in this sub-space. Because to meet multiplication closed
- All subspaces should include $ \ mathbf {0} $ vector
Therefore, there can be three-dimensional space of four sub:
- Straight line through the origin
- Plane through the origin
- Three-dimensional space itself
- 0 Vector
In summary therefore, the vector space is a set of vectors, the subspace is a subset of the set, regardless of the vector space or subspace, meet several closure by vector addition of
Third, the column space of a matrix
Suppose a matrix:
$A=\left[\begin{array}{lll}{1} & {1} & {2} \\ {2} & {1} & {3} \\ {3} & {1} & {4} \\ {4} & {1} & {5}\end{array}\right]$
Column space $ C (A) is the matrix $ $ \ mathrm {R} ^ {4} $ subspace, then the $ C (A) $ in the end? In fact, $ C (A) $ is a linear combination of matrix $ A $ in the column, then the column space of the matrix in the end what role?
Below we will contact the column space of linear equations up to a better understanding of $ Ax = b $, first $ Ax = b $ does not have a solution for all $ b $, because the combination of three vectors can not cover the entire 4-dimensional space, then what kind of $ b $ will make the equation has a solution it?
First, it is clear that when the equation class vector is zero when $ b $: $ \ left [\ begin {array} {l} {0} \\ {0} \\ {0} \ end {array} \ right] $, whenever b is zero vector equation solvability
In fact, think about it only when we know that $ b $ is the only solution to this equation when a linear combination of the columns of $ A $, that is to say if and only if $ b $ belong to $ A $ column space $ C (A) $ when $ Ax = b $ only solution
Therefore, the column space of the matrix is very important because it tells us what time solvable equation.