consider the basis S={v1,v2} for R^2,where v1=(-2,1) and
v2=(1,3),and let T:R^2-R^3 be linear transformation such that...
consider the basis S={v1,v2} for R^2,where v1=(-2,1) and
v2=(1,3),and let T:R^2-R^3 be linear transformation such that
T(v1)=(-1,2,0) And T(v2)=(0,-3,5), find T(2,-3)
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...
Find the standard matrix for the following transformation T : R
4 → R 3 :...
Find the standard matrix for the following transformation T : R
4 → R 3 : T(x1, x2, x3, x4) = (x1 − x2 + x3 − 3x4, x1 − x2 + 2x3 +
4x4, 2x1 − 2x2 + x3 + 5x4) (a) Compute T(~e1), T(~e2), T(~e3), and
T(~e4). (b) Find an equation in vector form for the set of vectors
~x ∈ R 4 such that T(~x) = (−1, −4, 1). (c) What is the range of
T?
Problem 2. (20 pts.) show that T is a linear transformation by
finding a matrix that...
Problem 2. (20 pts.) show that T is a linear transformation by
finding a matrix that implements the mapping. Note that x1, x2, ...
are not vectors but are entries in vectors. (a) T(x1, x2, x3, x4) =
(0, x1 + x2, x2 + x3, x3 + x4) (b) T(x1, x2, x3, x4) = 2x1 + 3x3 −
4x4 (T : R 4 → R)
Problem 3. (20 pts.) Which of the following statements are true
about the transformation matrix...
Problem 2. Show that T is a linear transformation by finding a
matrix that implements the...
Problem 2. Show that T is a linear transformation by finding a
matrix that implements the mapping. Note that x1,
x2, ... are not vectors but are entries in vectors.
(a) T(x1, x2, x3,
x4) = (0, x1 + x2, x2 +
x3, x3 + x4)
(b) T(x1, x2, x3,
x4) = 2x1 + 3x3 − 4x4
(T : R4 → R)
Please show T is a linear transformation for part (a) and
(b).
Topic: Math - Linear Algebra
Focus: Matrices, Linear Independence and Linear Dependence
Consider four vectors v1...
Topic: Math - Linear Algebra
Focus: Matrices, Linear Independence and Linear Dependence
Consider four vectors v1 = [1,1,1,1], v2 = [-1,0,1,2], v3 =
[a,1,0,b], and v4 = [3,2,a+b,0], where a and b are parameters. Find
all conditions on the values of a and b (if any) for which:
1. The number of linearly independent vectors in this collection
is 1.
2. The number of linearly independent vectors in this collection
is 2.
3. The number of linearly independent vectors in...