91Ó°ÊÓ

A vector base is a fundamental concept in linear algebra. It provides a way to represent any vector within a vector space uniquely as a linear combination of basis vectors. This concept allows us to understand the structure of vector spaces and perform operations on vectors.
A base for a vector space is a set of vectors that is both linearly independent and spans the entire space. "Linearly independent" means that no vector in the base can be expressed as a linear combination of the others. "Spans the space" implies that any vector in the vector space can be written as a combination of these basis vectors.

• For example, in two-dimensional space, the standard basis vectors are typically denoted as \( \mathbf{i} \) and \( \mathbf{j} \). In the three-dimensional case, it's \( \mathbf{i} \), \( \mathbf{j} \), and \( \mathbf{k} \).
• In our exercise, the sets \( \mathcal{D} = \{ \mathbf{d}_1, \mathbf{d}_2 \} \) and \( \mathcal{B} = \{ \mathbf{b}_1, \mathbf{b}_2 \} \) serve as bases for their respective vector spaces \( V \) and \( W \).

By studying and manipulating bases, we can simplify complex vector transformations, such as finding the matrix representation of a linear transformation.

Matrix representation is a powerful tool in linear algebra. It allows us to express linear transformations in a compact form that's easy to work with mathematically. A matrix essentially encodes the action of a linear transformation on certain bases of vector spaces.
To find a matrix representation for a linear transformation \( T \) between vector spaces, we determine how \( T \) acts on the basis vectors of the domain space and express those images in terms of the basis vectors of the codomain space. This results in a matrix where each column represents the image of a basis vector under the transformation:

• The transformation matrix \( [T]_{\mathcal{D}\mathcal{B}} \) is formed by stacking these column vectors.
• Given \( T(\mathbf{d}_1) = 2 \mathbf{b}_1 - 3 \mathbf{b}_2 \), the first column becomes \( \begin{bmatrix} 2 \ -3 \end{bmatrix} \).
• For \( T(\mathbf{d}_2) = -4 \mathbf{b}_1 + 5 \mathbf{b}_2 \), the second column is \( \begin{bmatrix} -4 \ 5 \end{bmatrix} \).
• Thus, the matrix \( [T]_{\mathcal{D}\mathcal{B}} \) is \( \begin{bmatrix} 2 & -4 \ -3 & 5 \end{bmatrix} \).

This matrix representation is crucial for simplifying calculations and visualizing linear transformations.

Linear algebra involves the study of vectors, vector spaces, and linear transformations, and is a foundational subject in many areas of mathematics and applied sciences. Here are some core concepts:
**Linear Transformation**: A function \( T: V \to W \) that maps between vector spaces and preserves vector addition and scalar multiplication. If \( T \) is linear, then for any vectors \( \mathbf{u}, \mathbf{v} \) and a scalar \( c \):

• \( T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v}) \)
• \( T(c\mathbf{u}) = cT(\mathbf{u}) \)

**Basis and Dimension**: The basis is a minimal set of vectors that spans a vector space, while dimension counts the number of vectors in that basis. The concepts help us understand the "size" and nature of vector spaces.
Linear transformations can often be captured and manipulated using matrices, which make computation easier.

• In the exercise, understanding how the transformation \( T \) acts on the basis helps construct its matrix representation.
• This highlights the efficiency and power of matrices in capturing systematic linear mappings.

Mastery of these concepts allows clearer insights into many mathematical and practical problems.