An Overview of Linear Algebra

 

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인공지능 전공자라면 정말 중요하지만 어렵게 느끼는 선형대수학 입니다.

 

오늘은 MIT GilbertStrang 교수님의 2강인 An Overview of Linear Algebra 에 대해 리뷰해보겠습니다.

 

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This is an overview of linear algebra given at the start of a course on the math­ ematics of engineering.

 

 

Vectors

What do we do with vectors? Take combinations

We can multiply vectors by scalars(such as under x1,x2,x3), add, and subtract. Given vectors u, v

and w we can form the linear combination x1u + x2v + x3w = b.

The collection of all multiples of u forms a line through the origin. The collec­ tion of all multiples of v forms another line. The collection of all combinations of u and v forms a plane. Taking all combinations of some vectors creates a subspace.

We could continue like this, or we can use a matrix to add in all multiples of w.

 

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Matrices

Create a matrix A with vectors u, v and w in its columns:

A times x very important of matrix times a vector.

This particular matrix A is a dif­ ference matrix because the components of Ax are differences of the components of that vector.

When we say x1u + x2v + x3w = b we’re thinking about multiplying num­ bers by vectors; when we say Ax = b we’re thinking about multiplying a matrix (whose columns are u, v and w) by the numbers. The calculations are the same, but our perspective has changed.

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For any input vector x, the output of the operation “multiplication by A” is some vector b

A deeper question is to start with a vector b and ask “for what vectors x does Ax = b?” In our example, this means solving three equations in three un­ knowns.

This is the solution.

It`s a matrix times b. Ax = b or x = A−1b. If the matrix A is invertible, we can multiply on both sides by A−1 to find the unique solution x to Ax = b.

 

Second example has the same columns u and v and replaces column vector w:

Our system of three equations in three unknowns becomes circular.

Where before Ax = 0 implied x = 0, there are non-zero vectors x for which Cx = 0. Foranyvectorxwithx1 = x2 = x3,Cx = 0. This is asignificant difference; we can’t multiply both sides of Cx = 0 by an inverse to find a non­ zero solution x.

 

The system of equations encoded in Cx = b is : (under picture)

Add these eaquations together, we get : (under picture)

This tells us that Cx = b has a solution x only when the components of b sum to 0.

In a physical system, this might tell us that the system is stable as long as the forces on it are balanced.

 

There are some troubles. So let`s see geometrically why were in trouble.

Geometrically, the columns of C lie in the same plane (they are dependent; the columns of A are independent).

Did not change u and v but I changed only w to minus one.

what does this mean?

Minus one sort of going zero, one is the z direction.

w star is a different w.

all combinations of u, v, and the third guy w*.

This matrix is not invertible, those three vectors are not a basis.

Their combinations are only in a plance(plane would be a typical subspace).

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Subspaces

Can we describe what vectors we get?

 

What b`s do we get? We do not get them all

We only get a plane of them. We get the ones where the components add to zero.

A vector space is a collection of vectors that is closed under linear combina­tions.

A subspace could be equal to the space it’s contained in; the smallest subspace contains only the zero vector.

 

 

What is the smallest subspace of R^3?

1. the origin

2. a line through the origin

3. a plane through the origin

4. all of R^3.

 

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Conclusion

When you look at a matrix, try to see “what is it doing?”

Matrices can be rectangular; we can have seven equations in three un­knowns.

Rectangular matrices are not invertible, but the symmetric, square matrix AT A that often appears when studying rectangular matrices may be invertible.

 

 

오늘은 MIT 길버트스트랭 교수님의 선형대수학 2강에 대해 리뷰해보았습니다.

 

다음에는 3강으로 리뷰해보겠습니다.