Linear Algebra Question, Vector Images

In summary, the conversation discusses finding a vector in the domain that maps to a given vector in the range of a linear transformation. The process involves setting up a system of linear equations and solving for the values of the vector. The conversation also mentions the use of a standard matrix to perform the transformation, but notes that the matrix cannot be inverted in this case.
  • #1
hemsley
2
0

Homework Statement


Let T:R^4->R^3 be the linear transformation de fined by

T( x1, x2,x3,x4) =
2(x1) - 4(x3)
(x2) -(x3)+3(x4)
(x1)+(x2)-3(x3)+2(x4)

Find the vector from the domain, Xd, which gives the image Xr = (2 1 1) in the range of T



The Attempt at a Solution


I don't need to necessarily know the values, I just need to know the process to get the 3 space matrix to find a vector in 4 space. To go from the Vector in Xd to the image Xr, I would just need to use the numbers in the matrix transform, or just multiply the standard matrix with the vector, but I am not sure how to go the other way. Do I take the inverse of the standard matrix multiply it by Xr possibly? Please help!
 
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  • #2
So you need to find [itex](x_1,x_2,x_3,x_4)[/itex] such that

[tex]T(x_1,x_2,x_3,x_4)=(2,1,1)[/tex]

Or

[tex](2x_1-4x_3,x_2-x_3+3x_4,x_1+x_2-2x_3+2x_4)=(2,1,1)[/tex]

Can you make a system of equations out of this??
 
  • #3
You need to use the the relation between the 4D vector (x1,x2,x3,x4) and the image vector (2,1,1):

2(x1) - 4(x3)=2
(x2) -(x3)+3(x4)=1
(x1)+(x2)-3(x3)+2(x4)=1.

This is a system of linear equations, solve it with some standard method. The matrix of the linear transformation is

T=
2 0 -4 0
0 1 1 3
1 1 -3 2

It can not be inverted.ehild

Edit: Micromass beat me ...
 
  • #4
Thanks very much to both of you!
 

Related to Linear Algebra Question, Vector Images

1. What is Linear Algebra?

Linear Algebra is a branch of mathematics that deals with linear equations, matrices, vectors, and vector spaces. It involves studying the properties and operations of these mathematical objects and their applications in various fields such as physics, engineering, and computer science.

2. What are vector images?

Vector images are graphics that are created using mathematical equations to define lines, curves, and shapes. They are composed of a series of points connected by lines and can be scaled to any size without losing quality.

3. How are vectors represented in Linear Algebra?

In Linear Algebra, vectors are typically represented as column matrices or coordinates in a coordinate system. They can also be represented graphically as arrows with a certain magnitude and direction.

4. What are the operations performed on vectors in Linear Algebra?

The main operations performed on vectors in Linear Algebra include addition, subtraction, scalar multiplication, dot product, and cross product. These operations have various applications in solving systems of linear equations, finding angles and distances, and representing physical quantities such as force and velocity.

5. What are the applications of Linear Algebra in real life?

Linear Algebra has numerous applications in various fields such as computer graphics, data analysis, cryptography, and machine learning. It is also used in engineering and physics to model and solve complex systems, and in economics and finance for optimization and risk analysis.

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