Linear algebra - transformations

In summary, to determine if the vector w is in the image (range) of a linear transformation T, you must find the matrix B that represents T and solve the system Bx = w. If the solution is consistent, then w is in the column space of B, which is the image (range) of T.
  • #1
Niles
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[SOLVED] Linear algebra - transformations

Homework Statement


Please take a look at:

http://www.math.luc.edu/~jdg/w3teaching/math_212/sp02/PDF/test2practice.pdf

Please take a look at #7, question c. To determine if the vector w is in the image (range) of T, I find the matrix B that represents the linear transformation T and find the solution to the system:

Bx = w,

because w has to be in the span of B (which I found to be the image of T). If it is consistent, w is in the range of T?

Thanks in advance,

sincerely Niles.
 
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  • #2
It's not clear to me what the "span" of a matrix would be! I think you mean the span of the vectors making the columns of the matrix T. Yes, that span, the "column space" of T is the image of the T and "image" is the "range" here. The crucial point is exactly how you show that w is in the image of T.
 
  • #3
I have the vectors that span the column space. I put these vectors together in a matrix I call B, and I write a new matrix <B|w>, and solve this. If consistent, w is in the column space.

Correct?
 

1. What are linear transformations?

Linear transformations are mathematical operations that map vectors from one space to another while preserving their basic structure. In other words, linear transformations are functions that take in a vector as an input and produce another vector as an output, where both vectors belong to the same vector space.

2. What are the properties of linear transformations?

There are three main properties of linear transformations: 1) linearity, 2) preservation of the origin, and 3) preservation of the scalar multiplication. Linearity means that the transformation follows the rules of addition and scalar multiplication. Preservation of the origin means that the zero vector in the input space is mapped to the zero vector in the output space. Preservation of the scalar multiplication means that multiplying a vector by a scalar in the input space will result in the same scalar multiplied by the vector in the output space.

3. How do linear transformations affect the shape of vectors?

Linear transformations can affect the shape of a vector in a variety of ways. They can stretch, shrink, rotate, reflect, and shear vectors. These transformations can also be combined to create more complex effects on the shape of a vector.

4. How are linear transformations represented?

Linear transformations can be represented in several ways, including matrices, equations, and geometrically. In matrix representation, each column of the matrix represents the transformation of the corresponding basis vector. In equation representation, the transformation is expressed as a function of the input vector. Geometrically, linear transformations can be represented as a change in the position, orientation, or size of a vector.

5. What is the importance of linear transformations in mathematics and science?

Linear transformations are essential in many areas of mathematics and science, including physics, engineering, computer graphics, and data analysis. They provide a powerful tool for understanding and manipulating vector spaces, which are fundamental in many mathematical and scientific concepts. Linear transformations also have practical applications in solving systems of linear equations, performing image and signal processing, and analyzing data sets.

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