Linear Algebra

Rank and solutions for consistent system of equations. If r = n, then the system has (1)______ solution(s). If r < n, then the system has (2)_______ solution(s)

(1) a unique, (2) Infinitely many

(A^-1)^-1 = ______

A

Linear combination

A sum of scalar multiples of vectors. The scalars are called the weights.

What is the set of all of the vectors that we can represent when we sum up a linear combination?

All real numbers in the two dimensional space

The rref(A) of an n x n matrix where every column is a linearly independent pivot column is known as _______

An identity matrix.

The only way that a transform matrix can be invertible is if the rref() of our transform matrix is equal to ______

An n x n identity matrix

(AB)^-1 = ______

B^-1 x A^-1

Linear Independence

Cannot write one of the vectors (in a subset/space) as a linear combination. And, in the general form of the plane/line --> t = 0.

If every column of an augmented matrix contains a pivot, then the corresponding system is _______ and has _______ solution(s)

Consistent, a unique

A system of equations is called homogeneous if _____

Each of the constant terms is equal to 0.

True or False. If a system of linear equations has 4 variables and 3 equations then it must have infinitely many solutions.

False, because the matrix could have the last row be like [0 0 0 0 | 1] which means it has no solution.

True or False. If a system of linear equations has 3 equations and 2 variables, then it has no solutions

False, the matrix could have a row of 0's meaning it has infinitely many solutions.

True or False. The matrix equation Ax = 0 has trivial solution if and only if there are no free variables.

False. If the last row in the matrix is something like [0 0 0 | 0], then there are free variables but still a solution. (Ax = 0 always has the trivial solution)

True or False. The zero vector 0 does not belong to Span(u,v) for u, v vectors in R^3.

False. Since 0u + 0V = [0 0 0]

True or False. If one of the rows of a 3x4 augmented matrix is equal to [0 1 0 4] then the associated system has a unique solution.

False. The matrix could also have a row equal to [0 0 0 0] which would give it infinitely many solutions, or a row [0 0 0 1] which is impossible so no solution.

True or false: suppose a homogenous system has the same number of variables as equation. Then the system has a unique solution

False. While is is possible for such a system to have a solution, it is also possible to have infinitely many. Remember homogeneous systems follow the rules that when rank = variables, yes there is a unique solution but when rank < variables there are infinitely many. Homogeneous solutions don't depend on the # of equations = variables. They care about rank

If a system in echelon form has a row of the form: [0 0 0 | b], b != 0 Then the system is _________ and has ______ solution(s)?

Inconsistent, no solution

If not all columns of the coefficient matrix of the echelon form are pivot columns, and the system is consistent, then the system has ______ solution(s).

Infinitely many

If the determinant of a matrix is not equal to zero than the matrix is ______

Invertible

Is is possible for a homogenous system to be inconsistent?

No, because there is always the trivial solution

Can any two vectors when summed represent all the dimensions in a 2D space? Explain.

No, when the two vectors are (collinear) multiples of each other. Meaning they are vectors on the same line. They only represent the span of that line.

If a system has a solution in which not all of the x_1....x_n are equal to 0, then we call this solution _______

Non-trivial

If the determinant of a matrix is equal to zero, then the matrix is ______

Not invertible

The rank of a system is the number of _______ in its echelon form

Pivot columns

What is the span of the zero vector?

The only vector I can get in the linear combination of a zero vector by itself, it's just the zero vector.

Linear Dependence

The set {v1,...,vp} is said to be linearly dependent if there exist weights c1,..,cp, not all zeros, such that c1v1 +...+cpvp = 0

If the rank (# of pivot columns) of a homogenous system is equal to the # of variables (n) then the system only has the [trivial |or| non-trivial] kind of solution. If r < n, then the system has ________ solutions. If the numbers of equations > n, the solution has ______ solutions.

Trivial, infinitely many, infinitely many

True or False. Let A be a 2 x 4 matrix, and B be a 5 x 4 matrix. A x B^T is defined and has dimensions 2 x 5.

True. A = 2 x 4, B^T transposes B into 4 x 5, when multiplying if the end of A and start of B^T are equal then we take the outside of each matrix and combine so 2 x 5.

True or False. If A is an augmented matrix of a system of linear equations such that all entries of the column of constants (the right most column) are equal to 0, then the system of equations represented by A is consistent.

True. Because even after row operations the RHS will still be 0, and in order to be inconsistent a row would have to equal [0 0 0 1] and since the RHS is == 0 that will never happen.

True or False. If A is a 4x5 matrix then A can have at most 4 pivot columns.

True. The max number of pivot columns a matrix can have is based on the amount of rows. Therefore a 4x5 matrix can have at most 4 pivot columns.

If A is invertible, Ax = b has a _______solution(s)

Unique

Is a homogenous system always consistent? Why or why not?

Yes because to be a homogenous system all equations must be equal to 0. Meaning they have a solution. Also known as the trivial solution.

In order for a matrix (A) to be invertible the rank(A) has to = ___ and ____. And everyone one of the columns is linearly __________.

m (# of rows) and n (# of columns). Meaning the matrix is a square matrix. Independent.