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Felicia E.

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What is a free variable?

When do they exist?

What happens if they exist?

For any value of the free variable, the other variables can be used to solve the equation. Free variables are found in the columns that have **no pivot elements**.

You have free variables if #equations**<**#unknowns

(Where unknowns is x_{1} x_{2} etc)

If you have free variables, the matrix has infinite solutions.

When does a linear system have infinite number of solutions?

- When it has free variables
- When it has a zero row
- When it has at least
**2 distinct solutions**.

When does a matrix have a **unique solution**?

- If #pivots
**=**#columns - If the determinant
**≠**0 - If a matrix is a square matrix and all of its columns are
**linearly independent**. - If it is invertible

i =

j =

m =

n =

i = top to bottom

j = left to right

m = rows

n = columns

When is AB (= crossproduct of A and B) defined?

When** #**columnsA **=** **#**rowsB

AxB is **not **BxA

A matrix is invertible if:

- AA
^{-1}= A^{-1}A = I - If you can turn it into the identity matrix by using row operations.
- detA ≠ 0
- Null A = {
__0__} --> for any b, the system has**at most 1**solution. - Col A = R
^{2}--> the system is**consistent for any b**.

The inverse is always **unique**.

If A is invertible, the solution to A__x__=__b__ is __x__=A^{-1}__b__

What is a singular matrix?

What is a regular matrix?

How can you check the inverse of a matrix?

A matrix is symmetric if:

What is a linear subspace?

What is the Null space?

How can you check if a vector is a linear combination of two other vectors.

How can you write the solution set of a homogeneous system?

And of a nonhomogeneous system?

A__x__=__b__ can only be a linear subspace of R^{n} if:

What is a column space?

How can you find it?

When is a set of vectors

- linear dependent?
- linear independent?

What is a basis?

How can you find it?

What do row operations affect?

When do you use column operations?

How can you determine of __b__ is in Col A?

detA =

What is detA of a triangular matrix?

How do elementary row operations affect the determinant?

Name 4 properties of the determinant:

What is the area of a parallelogram?

What is the volume of a parallelpipid?

What is a representation matrix?

What is Kernel?

What is image?

When is a transformation onto?

When one-to-one?

When is a transformation invertable?

How to find the composition of linear transformations.

(G°F)(__u__)

Name 3 properties of the identity transformation:

What is the representation matrix of a rotation?

What is the representation matrix of a projection onto a line?

What is the representation matrix of a reflection through a line?

What is the representation matrix of scaling a vector?

When is A__x__=λ__x__?

How can you determine the eigenspace?

When is a matrix diagonizable?

What is D?

(The diagonalization of A)

What is P?

How can you check them?

What is the easiest way to verify if __v__ is an eigenvector of A?

What is the easiest way to verify that λ is an eigenvalue of A?

Is every vector in E_{λ} and eigenvector of A?

How can you find the eigenvector if you know λ?

What is the abc formula?

How can you check is __v__ is in NullA?

How many pivot elements must an echelon form of A have if the columns of A span R^{m}?

If all columns of A are linearly independent, what do you know about the number of solutions?

Is a basis unique?

How can you find D if you know P?

if λ is an eigenvalue of A with eigenvector __x__, then A^{k}__x__ =

If A^{k} = 0, then the eigenvalue =

If A is invertible, and λ is an eigenvalue of A, then the eigenvalue of A^{-1} =

If T is invertible, then:

When is A nnot a basis?

One-to-one if:

Onto if:

(AB)^{-1}=

v is in Null A if:

T is diagonizable if it has n distinct eigenvalues or:

F-image of x =

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