A linear transformation is a function that maps vectors in one vector space to vectors in another vector space, while preserving the linear structure of the original space. This means that the transformation must satisfy the properties of linearity, such as preserving vector addition and scalar multiplication.
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Linear transformations preserve the linear structure of vector spaces, meaning they preserve vector addition and scalar multiplication.
Every linear transformation can be represented by a matrix, and every matrix represents a linear transformation.
The kernel of a linear transformation is the set of all vectors in the domain that are mapped to the zero vector in the codomain.
The rank of a linear transformation is the dimension of the image (or range) of the transformation.
The null space of a linear transformation is the same as the kernel of the transformation.
Review Questions
Explain how a linear transformation preserves the linear structure of vector spaces.
A linear transformation $T: V \to W$ preserves the linear structure of vector spaces by satisfying the following properties: 1) $T(\vec{u} + \vec{v}) = T(\vec{u}) + T(\vec{v})$ for all $\vec{u}, \vec{v}$ in the domain $V$, and 2) $T(c\vec{v}) = cT(\vec{v})$ for all scalars $c$ and all $\vec{v}$ in the domain $V$. These properties ensure that the transformation maintains the additive and scalar multiplication structure of the original vector space $V$.
Describe the relationship between a linear transformation and its associated matrix.
Every linear transformation $T: V \to W$ can be represented by a matrix $A$, where $T(\vec{v}) = A\vec{v}$ for all $\vec{v}$ in the domain $V$. Conversely, every matrix $A$ represents a linear transformation $T(\vec{v}) = A\vec{v}$. This means that the properties of a linear transformation, such as its kernel, rank, and null space, can be studied by analyzing the corresponding matrix $A$.
Explain the significance of the kernel and rank of a linear transformation.
The kernel of a linear transformation $T: V \to W$ is the set of all vectors in the domain $V$ that are mapped to the zero vector in the codomain $W$. The dimension of the kernel is called the nullity of the transformation. The rank of a linear transformation is the dimension of its image (or range), which is the set of all vectors in the codomain $W$ that are the images of vectors in the domain $V$. The rank and nullity of a linear transformation are related by the Rank-Nullity Theorem, which states that the sum of the rank and nullity is equal to the dimension of the domain $V$. These properties are crucial for understanding the properties and behavior of a linear transformation.
A vector space is a set of objects (called vectors) that can be added together and multiplied by numbers, called scalars, in a way that satisfies certain properties.
A matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns, that can be used to represent a linear transformation.
Kernel: The kernel of a linear transformation is the set of all vectors in the domain that are mapped to the zero vector in the codomain.