Consider the following two ordered bases of R^2:
B={〈1,−1〉,〈2,−1〉}
C={〈1,1〉,〈1,2〉}.
Find the change of coordinates matrix...
Consider the following two ordered bases of R^2:
B={〈1,−1〉,〈2,−1〉}
C={〈1,1〉,〈1,2〉}.
Find the change of coordinates matrix from the basis B to the
basis C.
PC←B=?
Find the change of coordinates matrix from the basis C to the
basis B.
PB←C=?
Let B={(1,1,1),(4,−2,0),(0,−3,2)} and
B′={(1,0,0),(1,−2,1),(1,3,−1)} be two ordered bases for the vector
space V=R3. Find the transition...
Let B={(1,1,1),(4,−2,0),(0,−3,2)} and
B′={(1,0,0),(1,−2,1),(1,3,−1)} be two ordered bases for the vector
space V=R3. Find the transition matrix from B to B′.
Let V be a three-dimensional vector space with ordered basis B =
{u, v, w}.
Suppose...
Let V be a three-dimensional vector space with ordered basis B =
{u, v, w}.
Suppose that T is a linear transformation from V to itself and
T(u) = u + v,
T(v) = u, T(w) =
v.
1. Find the matrix of T relative to the ordered basis B.
2. A typical element of V looks like
au + bv +
cw, where a, b and c
are scalars. Find T(au +
bv + cw). Now
that you know...
In R^2, let u = (1,-1) and v = (1,2).
a) Show that (u,v) form a...
In R^2, let u = (1,-1) and v = (1,2).
a) Show that (u,v) form a basis. Call it B.
b) If we call x the coordinates along the canonical basis and y
the coordinates along the ordered B basis, find the matrix A such
that y = Ax.
1. Find the orthogonal projection of the matrix
[[3,2][4,5]] onto the space of diagonal 2x2 matrices...
1. Find the orthogonal projection of the matrix
[[3,2][4,5]] onto the space of diagonal 2x2 matrices of the form
lambda?I.
[[4.5,0][0,4.5]] [[5.5,0][0,5.5]] [[4,0][0,4]] [[3.5,0][0,3.5]] [[5,0][0,5]] [[1.5,0][0,1.5]]
2. Find the orthogonal projection of the matrix
[[2,1][2,6]] onto the space of symmetric 2x2 matrices of trace
0.
[[-1,3][3,1]] [[1.5,1][1,-1.5]] [[0,4][4,0]] [[3,3.5][3.5,-3]] [[0,1.5][1.5,0]] [[-2,1.5][1.5,2]] [[0.5,4.5][4.5,-0.5]] [[-1,6][6,1]] [[0,3.5][3.5,0]] [[-1.5,3.5][3.5,1.5]]
3. Find the orthogonal projection of the matrix
[[1,5][1,2]] onto the space of anti-symmetric 2x2
matrices.
[[0,-1] [1,0]] [[0,2] [-2,0]] [[0,-1.5]
[1.5,0]] [[0,2.5] [-2.5,0]] [[0,0]
[0,0]] [[0,-0.5] [0.5,0]] [[0,1] [-1,0]]
[[0,1.5] [-1.5,0]] [[0,-2.5]
[2.5,0]] [[0,0.5] [-0.5,0]]
4. Let p be the orthogonal projection of
u=[40,-9,91]T onto the...
In R^3 consider the following two bases B= { v1=(2,2,-3),
v2=(2,2,0), v3=(1,2,4)} and B' = {...
In R^3 consider the following two bases B= { v1=(2,2,-3),
v2=(2,2,0), v3=(1,2,4)} and B' = { w1= (1,0,2), w2=(2,1,2),
w3=(0,2, -2) }
a) Find the matrix associated to the change of basis from B to
B'.
b) If VB= (-1,3,0), then find VB'