Problem 15 - Differential Equations and Linear Algebra

Question

Determine whether the given set \(S\) of vectors is a basis for \(M_{m \times n}(\mathbb{R})\). $$ \begin{array}{l} m=n=2: S=\left\\{\left[\begin{array}{rr} -3 & 1 \\ 0 & 2 \end{array}\right],\left[\begin{array}{rr} 3 & -5 \\ 6 & 1 \end{array}\right]\right. \\ \left.\left[\begin{array}{rr} -1 & -2 \\ 1 & 0 \end{array}\right],\left[\begin{array}{rr} 0 & 3 \\ 1 & -4 \end{array}\right],\left[\begin{array}{rr} 6 & -2 \\ -3 & -4 \end{array}\right]\right\\} \end{array} $$

Step-by-Step Solution

Answer

After Gaussian elimination, we obtain the following reduced row echelon form: $$ \left[\begin{array}{ccccc} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ \end{array}\right] $$ Since we have a unique solution (all zeros for the coefficients of the linear combination), the given set \(S\) is linearly independent. #tag_title#Step 2: Check if the set spans \(M_{2 \times 2}(\mathbb{R})\)#tag_content# Now we have to check if the set spans \(M_{2 \times 2}(\mathbb{R})\). The dimension of \(M_{2 \times 2}(\mathbb{R})\) is 4, and we have 5 matrices in the set \(S\). Since the matrices are linearly independent, and there are more than 4 elements in the set, the set must span \(M_{2 \times 2}(\mathbb{R})\). #tag_title#Conclusion#tag_content# Since the set \(S\) is both linearly independent and spans \(M_{2 \times 2}(\mathbb{R})\), it forms a basis for \(M_{2 \times 2}(\mathbb{R})\).

Step 1: Check for linear independence

We will first check if the matrices in the set \(S\) are linearly independent. We can do this by considering them as columns of a single matrix and performing Gaussian elimination. The given set of matrices is: $$ S=\left\{\left[\begin{array}{rr} -3 & 1 \\ 0 & 2 \end{array}\right],\left[\begin{array}{rr} 3 & -5 \\ 6 & 1 \end{array}\right],\left[\begin{array}{rr} -1 & -2 \\ 1 & 0 \end{array}\right],\left[\begin{array}{rr} 0 & 3 \\ 1 & -4 \end{array}\right],\left[\begin{array}{rr} 6 & -2 \\ -3 & -4 \end{array}\right]\right\} $$ Now consider this set of matrices as columns of a single matrix and perform Gaussian elimination: $$ A = \left[\begin{array}{ccccc} -3 & 3 & -1 & 0 & 6 \\ 1 & -5 & -2 & 3 & -2 \\ 0 & 6 & 1 & 1 & -3 \\ 2 & 1 & 0 & -4 & -4 \\ \end{array}\right] $$

Step 2: Identify the algebraic structure

Determine the type of algebraic problem.

Step 3: Apply algebraic techniques

Use factoring, expanding, or systematic methods.

Step 4: Simplify and solve

Simplify expressions and solve for unknowns.

Step 5: State the result

Write the final answer.

Step 6: Conclude with the answer

Since the set \(S\) is both linearly independent and spans \(M_{2 \times 2}(\mathbb{R})\), it forms a basis for \(M_{2 \times 2}(\mathbb{R})\).

Key Concepts

Basis Determination

In linear algebra, determining a basis for a vector space is a crucial task. A basis provides a way to describe every element in the space using linear combinations of the basis elements. To check if a set of vectors forms a basis, we need to ensure two key properties: linear independence and spanning the space.

In the context of matrices, specifically for the space of all real-valued matrices of size \(m \times n\), the set must span the whole space and have exactly \(m \times n\) linearly independent vectors. The dimension of the space of \(2 \times 2\) matrices is 4, which means a basis should also have 4 vectors. Therefore, we'll examine if the given set of matrices in our exercise can serve as a basis by checking these properties one by one.

Linear Independence

Linear independence is a fundamental concept in determining a basis. A set of vectors is linearly independent if no vector in the set is a linear combination of the others. To check for linear independence, we arrange the given matrices as columns of a larger matrix, then use Gaussian elimination to row-reduce it.

When the matrix is in row-echelon form, we look for pivot columns (columns with leading 1s). If each original vector corresponds to a pivot column and there are no free variables, the vectors are linearly independent.

In our exercise, the matrix representation of the set could be row-reduced to check for independence. If the number of pivot columns equals the number of vectors, they're independent. If not, the set is dependent, and thus it doesn't form a basis by itself.

Matrix Representation

When dealing with matrices as vectors, representing them correctly is crucial. In our exercise, each small matrix is seen as a vector by stacking its columns, which helps in forming the larger matrix for Gaussian elimination. This process aids in evaluating properties like independence and determining whether they span the space.

Each \(2 \times 2\) matrix has 4 entries, so each must be transformed into a column with 4 components. The five given matrices in this exercise formed columns of a \(4 \times 5\) matrix. This representation is essential for applying transformations and deciding about linear independence and span within the vector space.

Gaussian Elimination

Gaussian elimination is an algorithmic technique used to solve systems of linear equations. It's widely applied to determine the rank and verify the independence of a set of vectors in a matrix. The process involves three types of row operations:

• Swapping two rows.
• Multiplying a row by a non-zero scalar.
• Adding a multiple of one row to another row.

By applying these operations, the matrix is transformed into row-echelon form where the leading entries (also known as pivots) are easily identifiable.

For basis determination, the goal is to see if the matrix has pivots in all rows. In our task, using Gaussian elimination on the \(4 \times 5\) matrix helps us decide if the arranged columns are independent or span the entire space. If there are fewer than 4 pivots, it means the set does not span the space, and thus can't fully represent it as a basis.