Linear Algebra Final

Linear independence only has the ____________ solution

Trivial (o), where c = 0

A pivot in every column means that the set is _________ _______. This also meas there are NO ________ ______ and there is no _______ vector

linearly independent vectors, free variable , 0

How many pivots are in a line, plane, and a 3D space?

1, 2, 3

For something to be one:one, it must be…

every b in Rm has to have only 0 or 1 solution in Rn, A(x) = 0 has only the trivial solution (linealry independent) and col A are linearly indepednent

How can you test linear indepednence?

Do RREF, check pivots

For one:one, T(u) = T(v), so u = v, why?

Because if T(u) = T(v), then the transformation does not map distinct inputs to the same output, ensuring that each input corresponds to a unique output.

Rn is the ____ and Rm is the _________

domain, codomain

For onto transformations, what must be true?

The range of T = codomain of T, T9x) = b has 1 or infinitely many solutions for every b in Rm, pivot in every row

For onto transformations, you must have a pivot in every _________, which means the span of colA = _______

row, Rm

For onto transformations, the dim(Range T) = Rank (A) = ___

m

For linear transformations, T(u+v) = _____ and T(cu) = _______, why?

T(u) + T(v), c (T(u)), becasue it must be linearly indepedent and be a linear combination

How would you check for a linear transformation?

Plug in 0 for x, you should get 0 back, and it must be linearly indepdendnt and a linear combination

What CANNOT be a linear transformation?

absolute values, cos/sin/xy/x², basically anything that cannot be a linear combination

For matrix multiplication, T is _________ and U is _______

Rn to Rm, Rp to Rm

The composition of T * U (x) is _____, where _____ is applied first and it flows from ________

T (U (x)), U, Rp to Rm

The composition os T * U MUST have a _________ of U and a __________ of T

codomain, domain (essentially that the output if U is the input of T)

In matrix multiplication, AB show that the number of _____ of B is equal to the # _____ of A

columns, rows

For matrix multiplication, Does AB = BA? Does AB = 0 even when A an B are not equal to 0? What about AB = AC, does B=C even when A = 0?

NO, YES, NO B cannot equal C,

Compositions of linear transformations involve T (u) = Av and U(x) = Bx, so thate T * U = UT, which makes AB =

BA

What are the 3 qualifications for a subspace?

1. must have 0 vector

2. v and u, u+v are in V

3. the scaled vector must not be outside of the set

Any matrix has 2 subspaces, Column space of A is ___, and the null space of A is the subspace of _______, consisting of ______

Rm, Rn, Ax = 0

How do you find the null space?

RREF, Parametric form, attached to X is the null space

If AB = In, and BA = In, then B = ________ and (AB)^-1 =______

A-1, B^-1A^-1

A^-1 = _____

1/deta (d -b

-c a)

How do you know if something is inveritible vs non invertible?

invertible, det A not equal 0 not invertible if deta = 0

True or false: linear transformations are invertible if and only if, they are one to one and onto, because it is a square matrix

TRUE

Rank A + Null A = ______

n

what is cc by 90 vs c by 90?

cc = {0 -1, 1 0] c = [0 1 , -1 0]

whats reflection about the y vs reflection about the x?

x = [1 0 , 0 -1] y = [-1 0, 0 1]

if you have U * T, what would the order to solve be?

T goes first on the right, then U on the left