Linear Algebra
what is the formula that determines if linear independence
a + B + c = 0 (multiply this by the Vs and see if they still equal 0 )
General form of determining spanning
dim(P3) = 4
what is the basis of column space
each column with a pivot
What is the basis of row space
each row with a pivot
how to determine 1-1 or onto based on rank
if rank = columns then 1-1 but if rank = rows then onto
How to determine if P in invertible
if the determinant is not 0
what is isomorphism
if the rank equals both the column and rows or its 1-1 or Onto
If P = 2, what is P^2 [3x3]
it is 4
What does nullity represent
null in the columns
If P =2, what is P^-1 [3x3]
1/2
If P =2, what is P+2 [3x3]
2
If P = 2, what is 2P [3x3]
2(2)(2)= 8
what is a 2x4 matrix where dim(Nul(f)) = 3
3 columns of 0 and one with a value
Show the image of T is not all of V by finding a matrix NOT in the image
set the matrix equal to a random matrix that does not make it work
why does a 2x4 matrix with dim(col(d)) = 3 not exist
the main reason is that there is not 3 columns with pivots in the rows since there are only 2 rows
How to find the rank of a matrix
the number of pivots
What is 1-1
this means that each columns has a pivot
what is onto
this means that each row has a pivot
How to find the RREF
this means that we find the reduced row echelon from
How to find the inverse
use the 100,010,001 on the other side
Determine what Matrices are Ker(t)
you have to set the matrix = 0 and then determine the values of a b c d
How to determine if B is a basis of P^3
you to need to find independence and a spanning set
(T/F) Let dim(V)=2 and S to V if |S| = 3 then S spans V
False S has too many dim to span V
Let v =() Find Vb
Find the RREF
What does it mean when you Extend S to a basis for R2x2
Get the 2x2 matrix and make it into 4x4 then add (10000)
If it ask to find the Dimension of U what do you do
Make sure that you out the vectors together and solve for the rank
(T/F) the vector v => is in the null space of A ==>
Multiply the A by the vector v and if it does not equal 0 ,0,0 it si wirng
how to compute the basis of a kernel
Set Ax=0 and then solve the entire linear setq
yB x B = y
This is the formula for the values LOL
(T/F) There exists an injective linear transformation R(8x11) -> P73(R)
This is wrong because (8x11) > dim(p73(r)) = 74
(T/F) Let V be a vector space. If S1 to S2 to V, and S1 is linearly dependent then S2 is linearly dependetn
True
If L: R 3x4 --> P8(R) is surjective, then the nullity of L is 3
True (3)(4)=12 to 9 then 12-9 =3
Extend ... to a basis of R2x2
You make the 2x2 into a 4x1 and then add the e1 e2 e3 e4, You then do RREF and get rid of pivot columns then make the rows in the 2x2 matric