Find the linear transformation T : ℝ3 → ℝ2 that has the values given below on the given basis.T 1 00 = 3 8, T 1 10 = 2 6, T 1 11 = 7 5(a) First write a general vector v as a linear combination of the basis vectors.
Question
Find the linear transformation T : ℝ3 → ℝ2 that has the values given below on the given basis.T 1 00 = 3 8, T 1 10 = 2 6, T 1 11 = 7 5(a) First write a general vector v as a linear combination of the basis vectors.
Solution
The given basis vectors are (1,0,0), (0,1,0), and (0,0,1).
A general vector v in ℝ3 can be written as a linear combination of these basis vectors. Let's denote this vector as v = (x, y, z).
So, we can write v as:
v = x(1,0,0) + y(0,1,0) + z(0,0,1)
Now, we know how the linear transformation T acts on the basis vectors:
T(1,0,0) = (3,8) T(0,1,0) = (2,6) T(0,0,1) = (7,5)
So, we can write the action of T on v as:
T(v) = T(x(1,0,0) + y(0,1,0) + z(0,0,1))
Using the linearity of T (which means T(ax + by) = aT(x) + bT(y) for any vectors x and y and scalars a and b), we can write this as:
T(v) = xT(1,0,0) + yT(0,1,0) + zT(0,0,1)
Substituting the values we know for T acting on the basis vectors, we get:
T(v) = x(3,8) + y(2,6) + z(7,5)
So, the linear transformation T takes a vector (x, y, z) in ℝ3 and maps it to the vector (3x + 2y + 7z, 8x + 6y + 5z) in ℝ2.
Similar Questions
Find the linear transformation T : ℝ3 → ℝ2 that has the values given below on the given basis.T 1 00 = 3 8, T 1 10 = 2 6, T 1 11 = 7 5(a) First write a general vector v as a linear combination of the basis vectors.
Find the linear transformation T : ℝ2 → ℝ2 that has the values given below on the given basis.T 2 1 = 4 6, T 1 1 = 3 7(a) First write a general vector v as a linear combination of the basis vectors.v =v1 v2 = v1−v2 2 1 + 2v2−v1 1 1(b) This implies the following.T(v) = T v1 v2= 2v1 v1+6v2 = v
Find the matrix representation of the linear transformation T : R3 → R2 givenby the mapxyz 7 →( x + yx + z), where B =111 ,110 ,100and D ={( 10),( 02)}are bases for R3 and R2 respectively
Find a linear transformation 7: R2 R2 such that 7(1,-2) = (1, 1) and T(0, 1) = (2,3), where B = {(1,-2), (0, 1)) is a basis of IR2.
[5marks] letT:C3→C3bealinearmapsuchthat T(v1) = 2v1, T(v2) = v1 + 2v2, T(v3) = v2 + 2v3. A) Find the matrix T , i.e., the matrix of the linear map T with respect to B,B the basis B in the domain and the basis B in the co-domain. B) Find Te2 . B C) Find Te2.
Upgrade your grade with Knowee
Get personalized homework help. Review tough concepts in more detail, or go deeper into your topic by exploring other relevant questions.