Coordinates Flashcards
Given a basis B = (v1,v2,…,vm) of a subspace V of R^n, define the B-coordinates of a vector x in V and the corresponding B-coordinate vector.
Since x is in V, x can be written as:
x = c1v1 + c2v2 +…+ cmvm
The scalars c1,c2,…,cm are called the B-coordinates of x, and the B coordinate vector of x, denoted by [x]_B is . Note that x=S[x]_B, where S=
What two properties hold if B is a basis of a subspace V of R^n?
- [x+y]_B = [x]_B+[y]_B for all vectors x and y in V
2. [kx]_B=k[x]_B for all x in V and for all scalars k
Prove [kx]_B = k[x]_B
[kx]_B = = k = k[x]_B
Given v1, v2, and x, how do you find the B coordinate vector?
Find the scalars c1 and c2 that satisfy the following equation
x = c1v1 + c2v2
or
Create the matrix S by concatenating v1 and v2 and solve with the known equation x=S[x]_B
Describe the column approach to finding the similar matrix B to A.
B is equivalent to [[Av1],[Av2]…[Avm]]. Therefore you must multiply matrix A and each vector in (weird notation B) to find each column of the similar matrix
Describe how you can obtain a similar matrix B from a given matrix A and vectors that span (weird notation B)
S is comprised of the vectors that span (weird notation B).
B = S^-1AS
or AS = SB
What makes two matrices similar?
Consider two n x n matrices A and B. We say that A is similar to B if there exists an invertible matrix S such that
AS = SB or B=S^-1AS
Show that if matrix A is similar to B, then its power A^t is similar to B^t, for all positive integers t.
We know that B = S^-1AS for some invertible matrix S. Now
B^t = (S^-1AS)(S^-1AS)…(S^-1AS) = S^-1A^tS
Note the cancellation of many terms of the form S^-1S
Properties of similar matrices
- A n x n matrix A is similar to A itself (reflexivity)
- If A is similar to B, then B is similar to A (symmetry)
- If A is similar to B and B is similar to C, then A is similar to C (transitivity)
