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 \(A \vec{u}=A \vec{v}=\vec{b}\). $$ \begin{array}{l} A=\left[\begin{array}{cccc} 0 & -3 & -1 & -3 \\ -4 & 2 & -3 & 5 \end{array}\right], \\ \vec{b}=\left[\begin{array}{c} 0 \\ 0 \end{array}\right], \vec{u}=\left[\begin{array}{c} 11 \\ 4 \\ -12 \\ 0 \end{array}\right], \\ \vec{v}=\left[\begin{array}{c} 9 \\ -12 \\ 0 \\ 12 \end{array}\right] \end{array} $$

Short answer

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

Step-by-step solution

Understand the Problem

We need to verify that both vectors \(\vec{u}\) and \(\vec{v}\) satisfy the equation \(A\vec{x} = \vec{b}\). This means we must show that multiplying matrix \(A\) with each vector \(\vec{u}\) and \(\vec{v}\) results in vector \(\vec{b}\).

Matrix-Vector Multiplication for \(\vec{u}\)

Calculate \(A\vec{u}\). This involves the dot product of each row of matrix \(A\) with the vector \(\vec{u}\).For row 1 of \(A\):\[0(11) + (-3)(4) + (-1)(-12) + (-3)(0) = 0 - 12 + 12 + 0 = 0\]For row 2 of \(A\):\[(-4)(11) + 2(4) + (-3)(-12) + 5(0) = -44 + 8 + 36 + 0 = 0\]The result is:\[A\vec{u} = \begin{bmatrix} 0 \ 0 \end{bmatrix} = \vec{b}\]

Matrix-Vector Multiplication for \(\vec{v}\)

Calculate \(A\vec{v}\). Again, compute the dot product of each row of matrix \(A\) with the vector \(\vec{v}\).For row 1 of \(A\):\[0(9) + (-3)(-12) + (-1)(0) + (-3)(12) = 0 + 36 + 0 - 36 = 0\]For row 2 of \(A\):\[(-4)(9) + 2(-12) + (-3)(0) + 5(12) = -36 - 24 + 0 + 60 = 0\]The result is:\[A\vec{v} = \begin{bmatrix} 0 \ 0 \end{bmatrix} = \vec{b}\]

Conclusion

Since both vector multiplications \(A\vec{u}\) and \(A\vec{v}\) equal \(\vec{b}\), both \(\vec{u}\) and \(\vec{v}\) are solutions to the equation \(A\vec{x} = \vec{b}\).

Key concepts

Matrix-Vector Multiplication

Matrix-vector multiplication is an essential operation in linear algebra that combines a matrix and a vector to produce another vector. This process can be thought of as taking a weighted sum of the columns of the matrix, where the weights come from the vector.
Each element in the resulting vector is derived from performing a dot product between a row of the matrix and the vector.Here's how it works:

• Consider a matrix \(A\) with dimensions \(m \times n\) (\(m\) rows, \(n\) columns), and a vector \(\vec{x}\) with \(n\) elements.
• The multiplication \(A\vec{x}\) is defined if the number of columns in matrix \(A\) matches the number of elements in vector \(\vec{x}\).
• Each element in the resulting vector is calculated by taking the dot product of a row from matrix \(A\) and the vector \(\vec{x}\).

In the provided exercise, vectors \(\vec{u}\) and \(\vec{v}\) are each tested against matrix \(A\) to see if they satisfy the equation \(A\vec{x} = \vec{b}\). This involves calculating \(A\vec{u}\) and \(A\vec{v}\) step by step by performing dot products for each row of \(A\) with the given vectors. Both results are the zero vector \(\vec{b}\), confirming \(\vec{u}\) and \(\vec{v}\) as solutions.

Dot Product

The dot product, also known as the scalar product, is a fundamental operation in linear algebra, particularly in the context of matrix-vector multiplication. It is an operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number.To compute the dot product:

• Take two sequences of numbers, often represented as vectors.
• Multiply their corresponding components together.
• Sum up all the products to get the resulting scalar.

For example, to compute the dot product of two vectors \(\vec{a} = [a_1, a_2, ..., a_n]\) and \(\vec{b} = [b_1, b_2, ..., b_n]\), you calculate: \[\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 + ... + a_nb_n\]This operation is pervasive in the calculation of matrix-vector products because each element of the resulting vector is obtained by computing the dot product between a row of the matrix and the vector. In our exercise, we calculated the dot product for each row of the matrix \(A\) with vectors \(\vec{u}\) and \(\vec{v}\) to ensure that the equation results in \(\vec{b}\).

Linear Algebra

Linear Algebra is the branch of mathematics concerning linear equations, linear functions, and their representations through matrices and vector spaces. It has wide applications in various fields like computer science, physics, engineering, and economics. Some key aspects of linear algebra include:

• Understanding the behavior of linear transformations, which can be represented using matrices.
• Solving systems of linear equations, often through matrix operations including row reduction and matrix inversion.
• Studying vector spaces and subspaces which provide the framework for understanding linear equations.

The exercise highlights the intersection of these concepts, where matrix equations like \(A\vec{x} = \vec{b}\) are solved by determining if vectors satisfy the equation through calculated transformations using matrix-vector multiplication. By confirming both \(\vec{u}\) and \(\vec{v}\) are valid solutions, the fundamental principles of linear transformation and vector space properties are showcased, exemplifying linear algebra's role in such problems.