spratleyj said:Homework Statement
Show that if [tex]{ v_1, ... , v_k} [/tex] spans [tex] V [/tex] then [tex] {T(v_1), ... , T(v_k)}[/tex] spans [tex] T(v) [/tex]
we only need to show that [tex] {T(v_1), ... , T(v_k)}[/tex] spans [tex] T(c_1v_1 + ... + c_kv_k) [/tex]
A linear transformation is a mathematical operation that takes in one vector and produces another vector. It is a type of function that preserves the basic properties of vectors, such as direction and length.
A linear transformation can be represented by a matrix. The columns of the matrix correspond to the transformed basis vectors, and the values in each column represent how much the basis vector is scaled or rotated.
A basis is a set of linearly independent vectors that can be used to represent all other vectors in a vector space. It is the fundamental building block for understanding linear algebra and is essential for performing linear transformations.
A set of vectors is a basis if they are linearly independent, meaning that no vector in the set can be written as a linear combination of the other vectors. Additionally, the vectors must span the entire vector space, meaning that any vector in the space can be written as a linear combination of the basis vectors.
Yes, a linear transformation can change the dimension of a vector space. For example, a transformation that projects a 3-dimensional space onto a 2-dimensional plane will result in a lower dimension vector space. Similarly, a transformation that adds a new dimension to a vector space, such as a rotation in 2-dimensional space, will increase the dimension of the space.