Problem 21 - Differential Equations and Linear Algebra

Question

Determine the matrix representation \([T]_{B}^{C}\) for the given linear transformation \(T\) and ordered bases \(B\) and \(C,\) and for the given vector \(\mathbf{v}\) in \(V,\) use Theorem 6.5 .4 to compute \(T(\mathbf{v})\) \(T: \mathbb{R}^{3} \rightarrow M_{2}(\mathbb{R})\) given by $$T(a, b, c)=\left[\begin{array}{cc} 0 & -a+b+3 c \\ 5 a-c & -3 b \end{array}\right]$$ with \(B=\\{(0,1,0),(0,0,1),(1,0,0)\\}\) \(C=\left\\{E_{21}, E_{22}, E_{11}, E_{12}\right\\},\) and \(\mathbf{v}=(-2,1,-2)\)

Step-by-Step Solution

Answer

In summary, the matrix representation $[T]_{B}^{C}$ of the given linear transformation T with ordered bases B and C is: $$[T]_{B}^{C}=\left[\begin{array}{ccc}0 & 3 & 5 \\1 & 0 & 0 \\0 & -1 & 0\\0 & 0 & -3\end{array}\right]$$ And, for the given vector $\mathbf{v}=(-2,1,-2)$, we find the transformed vector as: $$T(\mathbf{v})=\left[\begin{array}{cc}11 & -2 \\-1 & 3\end{array}\right]$$

Step 1: Compute Images of Basis Vectors

First, we need to compute the images of the basis vectors in B under the transformation T. Apply T on each basis vector: $$T(0,1,0)=\left[\begin{array}{cc}0 & 1 \\0 & -3\end{array}\right] = E_{12}-3E_{22}$$ $$T(0,0,1)=\left[\begin{array}{cc}3 & 0 \\-1 & 0\end{array}\right] = 3E_{11}-E_{21}$$ $$T(1,0,0)=\left[\begin{array}{cc}0 & 0 \\5 & 0\end{array}\right] = 5E_{21}$$

Step 2: Express Images in Terms of C

Now, we need to express the images of these basis vectors in terms of the basis C, which we have already done in step 1. Use the coefficients of the basis elements in C as matrix elements: $$[T]_{B}^{C}=\left[\begin{array}{ccc}0 & 3 & 5 \\1 & 0 & 0 \\0 & -1 & 0\\0 & 0 & -3\end{array}\right]$$

Step 3: Find v in Basis B

Now, we have to express vector v in terms of basis B: $$\mathbf{v} = -2(1,0,0) + 1(0,1,0) + (-2)(0,0,1)$$ Thus, its representation in terms of basis B is: $$[\mathbf{v}]_{B}=(-2, 1, -2)$$

Step 4: Compute T(v) by Theorem 6.5.4

Theorem 6.5.4 states that if we're given a linear transformation T, the matrix representation [T]_{B}^{C} , and vector v in terms of basis B, we can compute T(v) by calculating: $$[T(\mathbf{v})]_{C}=[T]_{B}^{C}[\mathbf{v}]_{B}$$ Now, multiply [T]_{B}^{C} by [\mathbf{v}]_{B}: $$[T(\mathbf{v})]_{C}=\left[\begin{array}{ccc}0 & 3 & 5 \\1 & 0 & 0 \\0 & -1 & 0\\0 & 0 & -3\end{array}\right]\left[\begin{array}{c}-2 \\ 1 \\ -2\end{array}\right]=\left(\begin{array}{c}11\\-2\\-1\\3\end{array}\right)$$

Step 5: Express T(v) in the Original Space

The result we obtained is the coordinates of T(v) in terms of the basis C. Now convert it to the original space: $$T(\mathbf{v}) = 11E_{11} - 2E_{12} - E_{21} + 3E_{22} = \left[\begin{array}{cc}11 & -2 \\-1 & 3\end{array}\right]$$ This gives us the final answer: $$T(\mathbf{v})=\left[\begin{array}{cc}11 & -2 \\-1 & 3\end{array}\right]$$

Key Concepts

Linear Transformation

In the realm of mathematics, a linear transformation is a function that maps a vector space to another while preserving the operations of addition and scalar multiplication. This concept is crucial for understanding how datasets can be transformed or adjusted. For a function to qualify as a linear transformation, two main properties must hold:

• Additivity: Given two vectors \(\mathbf{u}\) and \(\mathbf{v}\), a linear transformation \(T\) satisfies \(T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v})\).
• Homogeneity of degree 1: For any scalar \(c\) and vector \(\mathbf{v}\), it satisfies \(T(c \cdot \mathbf{v}) = c \cdot T(\mathbf{v})\).

These properties ensure that the structure and relationships within the data are maintained. A classic example, as seen in many exercises, is a transformation \(T: \mathbb{R}^{3} \rightarrow M_{2}(\mathbb{R})\) where vector input in three-dimensional space is mapped onto a matrix as output. This process highlights how transformations can adapt data formats while retaining intrinsic relationships.

Ordered Bases

An ordered basis in a vector space is a sequence of vectors that spans the entire space and is linearly independent. The importance of ordered bases comes into play because they give us a way of describing every vector in the space as a unique linear combination of basis vectors.
For instance, if we consider a vector space \(V\), any vector \(\mathbf{v}\) in \(V\) can be expressed as \(c_1 \mathbf{b_1} + c_2 \mathbf{b_2} + \cdots + c_n \mathbf{b_n}\), where \(\mathbf{b_1}, \mathbf{b_2}, \ldots, \mathbf{b_n}\) are the basis vectors and \(c_1, c_2, \ldots, c_n\) are the coefficients.

• Uniqueness: The representation of a vector regarding a particular basis is unique.
• Order Matters: Even with the same set of vectors, a different order will result in a different basis unless it simply reorders the axes.

This uniqueness is vital when we change the basis representation of a vector in calculations, as in the exercises involving matrix representation and transforming vectors like \((1, 0, 0)\) or \((0, 1, 0)\) into \(T(a, b, c)\).

Vector Spaces

Vector spaces form the foundation for linear algebra, encompassing sets of vectors where two operations, vector addition and scalar multiplication, are defined and satisfy certain conditions. These conditions make vector spaces incredibly versatile and fundamental to mathematics and applied sciences.

• Closure: For any vectors \(\mathbf{u}\) and \(\mathbf{v}\) within a space, \(\mathbf{u} + \mathbf{v}\) also lies in that space.
• Existence of Zero Vector: There's a zero vector \(\mathbf{0}\) such that \(\mathbf{v} + \mathbf{0} = \mathbf{v}\).
• Scalar Multiplication: If \(\mathbf{v}\) is in the vector space and \(c\) is a scalar, then \(c \cdot \mathbf{v}\) is also in the space.
• Associativity and Commutativity: Vector addition is commutative \((\mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u})\) and associative, as is scalar multiplication.

These properties make vector spaces ideal for a wide variety of mathematical tasks, such as the calculation involving the transformation \(T\) mapping \(\mathbb{R}^{3}\) into the matrix space \(M_{2}(\mathbb{R})\). Understanding vector spaces allows us to comprehend transformations, structural properties, and manipulations.

Matrix Multiplication

Matrix multiplication is a critical operation in linear algebra that involves two matrices to yield a new matrix. It is essential for applications in linear transformations, data transformations, and systems of equations. Performing matrix multiplication entails combining rows from the first matrix with columns of the second matrix. Here's how it works:

• The number of columns in the first matrix must equal the number of rows in the second matrix.
• The resulting matrix has dimensions corresponding to the number of rows of the first and number of columns of the second matrix.
• Each element in the resulting matrix is computed as the dot product of the corresponding row and column.

For example, in the problem given, multiplying the matrix representation \([T]_{B}^{C}\) with \([\mathbf{v}]_{B}\) involves such row-to-column dot product operations, resulting in coordinates in the basis \(C\). This process combines the linear transformation with the vector's representation in the ordered basis \(B\), transforming it into a new representation in another space. Understanding matrix multiplication is vital for working with multiple transformations and for accurately capturing the essence of such fundamental linear operations.