Question

A matrix \(A\) and vectors \(\vec{b}, \vec{u}\) and \(\vec{v}\) are given. Verify that \(\vec{u}\) and \(\vec{v}\) are both solutions to the equation \(A \vec{x}=\vec{b} ;\) that is, show that \(A \vec{u}=A \vec{v}=\vec{b}\). $$ \begin{array}{l} A=\left[\begin{array}{ll} 1 & 0 \\ 2 & 0 \end{array}\right] \\ \vec{b}=\left[\begin{array}{l} -3 \\ -6 \end{array}\right], \vec{u}=\left[\begin{array}{l} -3 \\ -1 \end{array}\right], \vec{v}=\left[\begin{array}{l} -3 \\ 59 \end{array}\right] \end{array} $$

Short answer

Both \(\vec{u}\) and \(\vec{v}\) are solutions to \(A\vec{x} = \vec{b}\).

Step-by-step solution

Calculate A\vec{u}

First, we need to verify if \(A\vec{u} = \vec{b}\). Calculate \(A\vec{u}\):\[A = \begin{bmatrix} 1 & 0 \ 2 & 0 \end{bmatrix}, \quad \vec{u} = \begin{bmatrix} -3 \ -1 \end{bmatrix}\]Using matrix multiplication:\[A\vec{u} = \begin{bmatrix} 1 & 0 \ 2 & 0 \end{bmatrix} \begin{bmatrix} -3 \ -1 \end{bmatrix} = \begin{bmatrix} 1(-3) + 0(-1) \ 2(-3) + 0(-1) \end{bmatrix} = \begin{bmatrix} -3 \ -6 \end{bmatrix}\]This is equal to \(\vec{b} = \begin{bmatrix} -3 \ -6 \end{bmatrix}\). Thus, \(A\vec{u} = \vec{b}\).

Calculate A\vec{v}

Next, verify if \(A\vec{v} = \vec{b}\).Calculate \(A\vec{v}\):\[A = \begin{bmatrix} 1 & 0 \ 2 & 0 \end{bmatrix}, \quad \vec{v} = \begin{bmatrix} -3 \ 59 \end{bmatrix}\]Using matrix multiplication:\[A\vec{v} = \begin{bmatrix} 1 & 0 \ 2 & 0 \end{bmatrix} \begin{bmatrix} -3 \ 59 \end{bmatrix} = \begin{bmatrix} 1(-3) + 0(59) \ 2(-3) + 0(59) \end{bmatrix} = \begin{bmatrix} -3 \ -6 \end{bmatrix}\]This is equal to \(\vec{b} = \begin{bmatrix} -3 \ -6 \end{bmatrix}\). Therefore, \(A\vec{v} = \vec{b}\).

Conclusion

We have shown that \(A\vec{u} = \vec{b}\) and \(A\vec{v} = \vec{b}\).Therefore, both \(\vec{u}\) and \(\vec{v}\) are solutions to the equation \(A\vec{x} = \vec{b}\).

Key concepts

Matrix Multiplication

Matrix multiplication is a core concept in linear algebra. It involves multiplying matrices with vectors or other matrices. The most important rule to remember is that the number of columns in the first matrix must match the number of rows in the second matrix for multiplication to be possible. When you multiply a matrix by a vector, each element of the resulting vector is a sum of products. Each element of a row in the matrix is multiplied by the corresponding element in the vector, and then summed up. For example, given the matrix \( A = \begin{bmatrix} 1 & 0 \ 2 & 0 \end{bmatrix} \) and vector \( \vec{u} = \begin{bmatrix} -3 \ -1 \end{bmatrix} \), matrix multiplication \( A\vec{u} \) is computed as follows:

• First row of \( A \): \(1(-3) + 0(-1) = -3\)
• Second row of \( A \): \(2(-3) + 0(-1) = -6\)

The result is a vector \( \begin{bmatrix} -3 \ -6 \end{bmatrix} \). This final vector represents the linear transformation applied by matrix \(A\) on vector \( \vec{u} \).
Matrix multiplication is widely used in various applications, such as computer graphics, economics, and data analysis.

Solution Verification

Solution verification in the context of matrix equations involves confirming that a given vector satisfies the equation. By substituting the vector into the equation and ensuring the results match the required outcome, you can verify that the vector is indeed a solution. In our example, we have shown this verification process with vectors \( \vec{u} \) and \( \vec{v} \) that must satisfy matrix equation \( A\vec{x} = \vec{b} \). To verify, we calculated \( A\vec{u} \) and \( A\vec{v} \) using matrix multiplication. Both computations resulted in \( \begin{bmatrix} -3 \ -6 \end{bmatrix} \), which matches vector \( \vec{b} \).

• For vector \( \vec{u} = \begin{bmatrix} -3 \ -1 \end{bmatrix} \), the operation proved \( A\vec{u} = \vec{b} \).
• For vector \( \vec{v} = \begin{bmatrix} -3 \ 59 \end{bmatrix} \), the operation also confirmed \( A\vec{v} = \vec{b} \).

If a solution verification procedure shows discrepancies, adjustments to the calculations or reevaluations of the potential solutions might be needed. Ensuring that vectors satisfy a matrix equation is crucial for accurately solving systems of linear equations.

Linear Algebra

Linear Algebra is the branch of mathematics dedicated to vectors, vector spaces, and linear mappings between spaces. It underpins many aspects of science and engineering, making it an essential area of study for solving systems of linear equations.
In our exercise, the problem revolves around demonstrating potential solutions to the matrix equation \(A\vec{x} = \vec{b}\). Such problems lie at the heart of linear algebra concepts. Linear transformations, represented as matrix equations, map input vectors (like \(\vec{u}\) and \(\vec{v}\)) to output vectors (like \(\vec{b}\)).
Important concepts in linear algebra include:

• Vector Spaces: Collections where vectors can be added together and multiplied by scalars.
• Linear Mappings: Functions that preserve vector addition and scalar multiplication.
• Matrices: Representations of linear maps between vector spaces.

Learning linear algebra opens doors to advanced topics like eigenvalues and eigenvectors, matrix factorizations, and solutions to differential equations. Grasping these concepts allows you to explore deeper mathematical fields and modern computational technologies.