Linear Algebra

Subspace

theres the 0 vector, x+y are closed under vector addition, cy are closed under scalar multiplication

Linear independency?

You find if its linearly indepdent by the trivial solution. Its linearly dependent if there is a nontrivial solution. You can find it by row reducing it and if it has a pivot then its a column that is linearly indepdent, if it has a leading variable, its linearly dependent

nullity

the number of free variables in null space because you find null(a) by using free variables

extraction theorem

basis

How a basis shows the geometry 1. The number of basis vectors tells you the dimension

This is the first big geometric clue.

• If the basis has 1 vector, the space is a line through the origin.

• If the basis has 2 vectors, the space is a plane through the origin.

• If the basis has 3 vectors, the space is all of R3\mathbb{R}^3

  R3

  .

In your example, S′S'

S′

has two independent vectors, so the subspace is a plane.

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2. The basis vectors describe the directions of the subspace

A basis gives the fundamental directions that generate the whole subspace.

For example:

S′={[10−1],[111]}S' = \left\{ \begin{bmatrix}1\\0\\-1\end{bmatrix}, \begin{bmatrix}1\\1\\1\end{bmatrix} \right\}

S′=⎩⎨⎧​​10−1​​,​111​​⎭⎬⎫​

These two vectors form the edges of a parallelogram. Any vector in the plane is made by combining these two.

So geometrically:

• One direction is (1,0,−1)(1,0,-1)

  (1,0,−1)

  .

• Another direction is (1,1,1)(1,1,1)

  (1,1,1)

  .

• Together they sweep out an infinite plane.

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3. A basis removes redundant vectors

The original spanning set had 4 vectors, but some were combinations of others. This makes it hard to visualize.

The basis strips it down to only the essential directions.

This makes the geometry easy: you only look at the basis vectors.

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Summary

A basis helps you understand geometry because:

1. Its size tells you the geometric object (line, plane, or space).

2. Its vectors tell you the directions inside that object.

3. It removes any extra clutter so the geometry becomes clear.

If you want, I can show how the plane equation comes from the basis, or draw a picture of these two directions.

nullity

n - rank(a)

dim(null(a))

nullity(a)

dim(col(a))

rank(a)

rank(a) + nullity(a)

n

show that something is not linear

you have to do t(x+y) = t(x) + t(y) and t(cx) = c(t(x))

when we divide complex numbers, what do we do?

multiply by the conjugate

For determinants, when you swap 2 rows or swap 2 columns,

the determinant swaps the sign (- to +) or vice versa

if you add a multiple of a row like r1 + 3r3 then the determinant

stays the same

if you multiply a row by a constant, the determinant

det(B) = cDet(A)

det(AB) = det(A)det(B) = det(B)det(A) = det(BA)

true

det(a^T) = what?

det(a)

det(a+b) = det(a) + det(b)

false