Problem 5 - Differential Equations and Linear Algebra

Question

(a) find \(n\) such that rowspace \((A)\) is a subspace of \(\mathbb{R}^{n}\), and determine a basis for rowspace \((A) ;\) (b) find \(m\) such that colspace \((A)\) is a subspace of \(\mathbb{R}^{m},\) and determine a basis for colspace \((A)\). $$A=\left[\begin{array}{llll} 1 & 1 & -3 & 2 \\ 3 & 4 & -11 & 7 \end{array}\right]$$

Step-by-Step Solution

Answer

(a) rowspace \((A)\) is a subspace of \(\mathbb{R}^2\) with a basis \(\begin{bmatrix} 1 & 1 & -3 & 2 \\ 0 & 1 & -2 & 1 \end{bmatrix}\) (b) colspace \((A)\) is a subspace of \(\mathbb{R}^2\) with a basis \(\begin{bmatrix} 1 & 3 \\ 0 & 1 \end{bmatrix}\)

Step 1: Write down the given matrix and its transpose

Given matrix \(A\) is $$A = \begin{bmatrix} 1 & 1 & -3 & 2 \\ 3 & 4 & -11 & 7 \end{bmatrix}$$ We also need to find the transpose of matrix \(A\), which can be used to determine the colspace \((A)\) later. $$A^T = \begin{bmatrix} 1 & 3 \\ 1 & 4 \\ -3 & -11 \\ 2 & 7 \end{bmatrix}$$

Step 2: Perform row reduction on the given matrix A

Now we will perform row reduction on matrix \(A\) to obtain its row-echelon form: 1. Multiply the first row by -3 and add it to the second row, then replace the second row with the result. $$A = \begin{bmatrix} 1 & 1 & -3 & 2 \\ 0 & 1 & -2 & 1 \end{bmatrix}$$ Now we've obtained the row-echelon form of matrix \(A\).

Step 3: Find the row space of A

The non-zero rows in the row-echelon form of matrix \(A\) provide a basis for the row space of A. $$\text{basis for rowspace}(A) = \begin{bmatrix} 1 & 1 & -3 & 2 \\ 0 & 1 & -2 & 1 \end{bmatrix}$$ Since there are two vectors in the basis, rowspace \((A)\) is a subspace of \(\mathbb{R}^n\) where \(n=2\).

Step 4: Perform row reduction on the transpose matrix A^T

Now we will perform row reduction on matrix \(A^T\) to obtain its row-echelon form: 1. Subtract the first row from the second row, then replace the second row with the result. 2. Add 3 times the first row to the third row, then replace the third row with the result. $$A^T = \begin{bmatrix} 1 & 3 \\ 0 & 1 \\ 0 & 0 \\ 0 & 1 \end{bmatrix}$$

Step 5: Find the column space of A and its basis

The non-zero rows in the row-echelon form of matrix \(A^T\) provide a basis for the colspace of A. $$\text{basis for colspace}(A) = \begin{bmatrix} 1 & 3 \\ 0 & 1 \end{bmatrix}$$ Since there are two vectors in the basis, colspace \((A)\) is a subspace of \(\mathbb{R}^m\) where \(m=2\). So now we have our final answers: (a) rowspace \((A)\) is a subspace of \(\mathbb{R}^2\) and its basis is \(\begin{bmatrix} 1 & 1 & -3 & 2 \\ 0 & 1 & -2 & 1 \end{bmatrix}\) (b) colspace \((A)\) is a subspace of \(\mathbb{R}^2\) and its basis is \(\begin{bmatrix} 1 & 3 \\ 0 & 1 \end{bmatrix}\)

Key Concepts

Row Space

The row space of a matrix is a fundamental concept in linear algebra that refers to the set of all possible linear combinations of its row vectors. Simply put, it tells us what outputs we can potentially create by using the rows of a matrix as building blocks.
To find the row space, perform row reduction on the matrix until it reaches a row-echelon form (or even a reduced row-echelon form if needed). Once in this form, the non-zero rows left in the matrix form a basis for the row space.
For example, in our exercise, the row space of matrix \(A\) is formed by the non-zero rows in its row-echelon form:

• \([1, 1, -3, 2]\)
• \([0, 1, -2, 1]\)

These vectors span a subspace of \(\mathbb{R}^n\) where \(n=4\) because the original matrix has 4 columns, but since our basis has 2 vectors, the row space is essentially a 2-dimensional subspace.

Column Space

Column space, just like row space, is another essential concept which deals with the span of the matrix columns. This space shows all possible vectors that can be obtained by scaling and adding the column vectors of a matrix.
To find the column space, you generally look at the pivot columns identified during the row reduction of the matrix transpose, \(A^T\). The corresponding columns in the original matrix \(A\) usually form the basis for its column space.
In our matrix \(A\), after transposing and reducing, we found that:

• \([1, 3]^T\)
• \([0, 1]^T\)

These are basis vectors for the column space indicating that it is a 2-dimensional subspace of \(\mathbb{R}^m\) where \(m=2\), which is derived from the 2 pivot positions after reduction.

Matrix Transposition

Matrix transposition is a simple yet powerful operation in linear algebra where you "flip" the matrix over its diagonal. This means swapping the row and column indices for the elements, converting rows of the original matrix into columns and vice versa.
The transpose of a matrix \(A\), denoted as \(A^T\), plays a crucial role, especially in determining the column space of the original matrix \(A\). By examining the row reduced form of \(A^T\), we efficiently find the basis of the column space of \(A\).
For the given matrix \(A\):

• Original matrix: \(A = \begin{bmatrix} 1 & 1 & -3 & 2 \ 3 & 4 & -11 & 7 \end{bmatrix}\)
• Its transpose: \(A^T = \begin{bmatrix} 1 & 3 \ 1 & 4 \ -3 & -11 \ 2 & 7 \end{bmatrix}\)

All operations that you perform on \(A^T\), such as row reduction, help reveal key properties about the original matrix \(A\).

Row Reduction

Row reduction, often referred to as Gaussian elimination, is a method used to simplify matrices to make solving linear equations easier. It systematically transforms a matrix into its row-echelon form, and sometimes further into reduced row-echelon form.
Row reduction involves three types of operations: exchanging two rows, multiplying a row by a non-zero scalar, and adding or subtracting multiples of one row from another. These operations help identify pivotal rows and columns, providing information about the row and column spaces.
In our exercise, we performed row reduction on:

• Matrix \(A\) to span the row space.
• Transpose \(A^T\) allowing us to find pivotal structures for column space.

This technique unveiled the row space basis \([1, 1, -3, 2], [0, 1, -2, 1]\) and the column space basis from \(A^T\) for \(A\), allowing us to understand its intrinsic dimensional properties.