Eigenspace
The solution space of is called the eigenspace of associated with the eigenvalue . The eigenspace is denoted by .

Sum/Product of Eigenvalues
– The sum of all eigenvalues of (including repeated eigenvalues) is the same as (trace of , i.e. the sum of diagonal elements of )
– The product of all eigenvalues of (including repeated eigenvalues) is the same as .

rotates points in the -plane counterclockwise through an angle about the origin.

For example rotating the vector 45 degrees counterclockwise gives us:

Finding Least Squares Solution

Given (inconsistent system), solve
instead to get a least squares solution of the original equation.

Projection

If we know a least squares solution of , we can find the projection of onto the column space of by

Dimension Theorem for Matrices (Also known as Rank-Nullity Theorem)

If is a matrix with columns, then

(=number of pivot columns,

=number of non-pivot columns.)

Linear Independence and the Wronskian
A set of vector functions from to is linearly independent in the interval if for at least one value of in the interval .

Row echelon form (REF)
For each non-zero row, the leading entry is to the right of the leading entry of the row above.

E.g.

Note that the leading entry 9 of the second row is to the right of the leading entry 1 of the first row.

Reduced row echelon form (RREF)
A row echelon form is said to be reduced, if in each of its pivot columns, the leading entry is 1 and all other entries are 0.

E.g.

Elementary Row Operations
1) — multiply the th row by the constant
2) — swap the th and the th row
3) — add times of the th row to the th row.

Gaussian Elimination Summary
Gaussian Elimination is essentially using the elementary row operations (in any order) to make the matrix to row echelon form.

Gauss-Jordan Elimination
After reaching row echelon form, continue to use elementary row operations to make the matrix to reduced row echelon form.

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