math 3a

How to determine if set is a Linear combination?

Solve the matrices and if you get a row with 0 answer is yes.

Matrix products

Amount of columns should be equal to the rows. Left to right, top to bottom

Equation consistent for all vectors in R3?

Only if there are 3 columns and 3 rows. If only 2, there would only be x1 and x2 which is left/right and up/down.

Linear transformation

L(cx) = cL(x)

L(x+y) = L(x) + L(y)

One to one

For any entry in range, there is at most one domain mapped to it

Onto

For any element in range there is at least one element domain mapped to it

Linear independence in one to one

One to one if columns of A are linearly independent. Meaning in c1v1+c2v2 etc coefficients of C all vanish.

How to tell if the lines have a common point of intersection?

Use reduced echelon form and the last row has to be all 0s.

Augmented Matrix

Augmented means the last column is what the rows are equal to, so in reality it is 3 variables and what they are equal to.

How to tell if vector b is a linear combination?

Solve the matrix using REF, if bottom row is all equal to 0, it guarantees a solution so it is a linear combination.

Vector space

Two operations, addition of vectors, multiplication of vectors by numbers

Linear independence example:

every column has a leading entry so yes.

compute product of matrices

3 rules of Vector subspace

1) any value of a that makes the collection p(t) = 0 for all of t?

2) is it closed under addition? (add 2 of the equations together, one with variable a and the other with b, the result should match the same format as original equation. ie. a+t² shows constant plus one t².

3) is it closed under scalar multiplication? multiply by c and if the format is not the same as original then it is not a subspace.

Vector Subspace problem:

1) plugging in values for a still leave t² so first rule fails. the collection will not be equal to 0, instead would be equal to t²

2) adding (a+t²) + (b+t²) = (a+b) + 2t² —> does not follow the original format of constant plus one t² , it instead has 2t²

3) multiplication by c: lets just say 5: 5(a+t²) = 5a+5t² —> does not follow original format.

Linear transformation example

only use the transformed values, the initial values are just distractions.

How to determine if a collection is a basis?

Must be linearly independent: must have leading entry for each column. (answer was no because REF showed 000 for the bottom row)

How to determine if vectors given constitute a basis?

Linear Independence, Spanning

It constitutes a basis if it is LINEARLY INDEPENDENT and span

Use reduced echelon form to make sure each column has a leading entry, or look at the amount of columns and rows. If # of columns does not equal # of rows then it is linearly dependent and is NOT a basis.

Understanding DImension

n amount of elements = dimension n

EX. 3 columns (vectors) = dimension 3

vector space is fixed, dimension n will have n elements.

Relationship between number of rows/columns of 2 matrices so that multiplication of matrices is defined?

Number of columns in first matrix = number of rows in the second matrix

what is needed to identify a vector in an abstract vector space with a column of numbers?

make a choice of basis, expand the vector as a linear combination of basis elements, and make a column of the coefficients.

Define Span