Under what conditions does there exist a linear function L: ℝ^2 →ℝ^2 such that L( x1 y

step 1 It is given that the matrix A is a 2 cross 2 matrix with entries x 0Y. Now the determinant of A

Under what conditions does there exist a linear function L:...

Under what conditions does there exist a linear function $L: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ such that $L\left(\begin{array}{l}x_1 \\ y_1\end{array}\right)=\left(\begin{array}{l}a_1 \\ b_1\end{array}\right)$ and $L\left(\begin{array}{l}x_2 \\ y_2\end{array}\right)=\left(\begin{array}{l}a_2 \\ b_2\end{array}\right)$ ? Under what conditions is $L$ uniquely defined? In the latter case, write down the matrix representation of $L$.

Key Concepts

Linear Transformations

A linear transformation is a function between vector spaces that preserves vector addition and scalar multiplication. This means that for a linear transformation L, the equation L(cx + dy) = cL(x) + dL(y) holds for any vectors x, y and scalars c, d. This property is fundamental in determining how the transformation behaves and how its action is uniquely specified by its effect on a basis of the domain.

Matrix Representation

Any linear transformation between finite-dimensional vector spaces can be represented as a matrix with respect to chosen bases for the domain and codomain. The columns of this matrix correspond to the images of the basis elements under the transformation, making the matrix representation a compact way to encode the function’s action on the entire space.

Basis and Linear Independence

In the context of linear maps, a basis of a vector space is a set of linearly independent vectors that span the space. If the vectors for which the images are prescribed in the domain form a basis, then the linear transformation is uniquely determined by those images. Linear independence is critical because if the given vectors are linearly dependent, they do not form a basis, and additional consistency conditions are required for the existence of a linear transformation with the prescribed images.

Consistency Conditions When Vectors Are Dependent

When the given domain vectors do not form a basis (i.e., they are linearly dependent), there must be a consistency condition between their images for a linear transformation to exist. Specifically, if one of the domain vectors can be expressed as a scalar multiple or a linear combination of the other, then the corresponding image must satisfy the same linear relationship. Without this consistency, no linear transformation can map the vectors as prescribed.

Please give Ace some feedback