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Chapter 2: Problem 49

Prove that the scalar 1 is the identity for scalar multiplication: \(1 A=A\)

Short Answer

Expert verified
Proving that the scalar 1 is the identity for scalar multiplication involves defining scalar multiplication and applying this definition to matrix A. By showing that \(1 \cdot A = A\), we have demonstrated that the scalar 1 serves as the mathematical identity for the operation of scalar multiplication.

Step by step solution

01

Define Scalar Multiplication

Begin by defining scalar multiplication: a scalar value multiplied by a matrix is just the scalar value multiplied by every entry in the matrix. We can write this in more mathematical terms: for a matrix \(A = [a_{ij}]\), and a scalar \(\lambda\), \(\lambda \cdot A = [\lambda \cdot a_{ij}]\).
02

Apply Definition with Scalar 1

Now that the scalar multiplication has been defined, apply it using 1 as the scalar: \(1 \cdot A = [1 \cdot a_{ij}]\). By definition of multiplication, multiplying any number by 1 gives the same number. Thus, we can simplify to \(1 \cdot A = [a_{ij}]\), which equals matrix A.
03

Conclusion

In conclusion, through defining scalar multiplication and applying it to scalar 1, we have shown that \(1 \cdot A = A\), demonstrating that 1 is indeed the identity for scalar multiplication.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Scalar Multiplication
When we talk about scalar multiplication in the context of linear algebra, we're referring to the process of multiplying each entry of a matrix by the same number, which is known as a scalar. In practical terms, if we have matrix A with entries a_{ij} and we multiply it by a scalar (lambda), here's what happens: the result is a new matrix where each entry is a_{ij} multiplied by (lambda).

So why is this operation important? Scalar multiplication is one of the foundational operations in linear algebra because it's integral to the concepts of scaling and transformations. For instance, when dealing with transformation of geometrical figures in a coordinate system, scalar multiplication allows us to easily enlarge or shrink the figures.
Matrix Operations
Beyond scalar multiplication, matrix operations encompass a variety of computations including addition, subtraction, and multiplication of matrices. These operations are fundamental in linear algebra and have strict rules that must be followed. For example, when adding or subtracting matrices, they must be of the same dimensions, whereas for matrix multiplication to be valid, the number of columns in the first matrix must match the number of rows in the second matrix.

Understanding how to properly perform these operations is crucial for students because matrices are not just arrays of numbers; they represent complex systems and transformations in higher-dimensional spaces. Being able to manipulate these matrices accurately allows for the solving of real-world problems in areas such as physics, economics, and engineering.
Identity Matrix

What Is an Identity Matrix?

Within the realm of matrices, there exists a special type called the identity matrix, which plays a similar role to the number 1 in regular multiplication. The identity matrix, typically denoted as I, has 1's on its diagonal (top left to bottom right) and 0's elsewhere. No matter what matrix A you multiply it by (assuming correct dimensions), matrix A remains unchanged.

Why Identity Matrices Matter

Identity matrices are significant because they are the neutral element in matrix multiplication, just as zero is the additive neutral element. They are crucial in concepts like solving systems of linear equations, finding matrix inverses, and understanding linear transformations. Learning about the identity matrix is key in having a deeper grasp of matrix algebra and its properties.
Linear Algebra Proofs
Proofs in linear algebra often seem daunting to students, but they serve a critical purpose: to rigorously verify the truths we assume within the subject. When tackling a linear algebra proof, we must start by understanding the definitions of the concepts involved and setting up the necessary knowns and unknowns. The exercise we've been discussing is an example of a proof that demonstrates a fundamental property of scalar multiplication.

Proofs not only confirm the properties of operations such as scalar multiplication but also deepen our understanding of the structure and consistencies within linear algebra. They build a solid foundation for further mathematical exploration and application, and help to develop analytical and logical reasoning skills, which are incredibly valuable in the field of mathematics and beyond.

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Most popular questions from this chapter

Is the sum of two elementary matrices always elementary? Explain why or why not and provide appropriate examples to illustrate your conclusion.

Express the column matrix b as a linear combination of the columns of \(A\) $$A=\left[\begin{array}{rr} -3 & 5 \\ 3 & 4 \\ 4 & -8 \end{array}\right], \quad \mathbf{b}=\left[\begin{array}{r} -22 \\ 4 \\ 32 \end{array}\right]$$

A fruit grower raises two crops, apples and peaches. Each of these crops is shipped to three different outlets. The number of units of crop \(i\) that are shipped to outlet \(j\) is represented by \(a_{i j}\) in the matrix $$A=\left[\begin{array}{lll} 125 & 100 & 75 \\ 100 & 175 & 125 \end{array}\right]$$ The profit per unit is represented by the matrix \\[ B=\left[\begin{array}{ll} \$ 3.50 & \$ 6.00 \end{array}\right] \\] Find the product \(B A\) and state what each entry of the product represents.

Prove that \(A\) is idempotent if and only if \(A^{T}\) is idempotent. Getting Started: The phrase "if and only if" means that you have to prove two statements: 1\. If \(A\) is idempotent, then \(A^{T}\) is idempotent. 2\. If \(A^{T}\) is idempotent, then \(A\) is idempotent. (i) Begin your proof of the first statement by assuming that \(A\) is idempotent. (ii) This means that \(A^{2}=A\) (iii) Use the properties of the transpose to show that \(A^{T}\) is idempotent. (iv) Begin your proof of the second statement by assuming that \(A^{T}\) is idempotent.

Factor the matrix \(A\) into a product of elementary matrices. $$A=\left[\begin{array}{lll} 1 & 2 & 3 \\ 2 & 5 & 6 \\ 1 & 3 & 4 \end{array}\right]$$
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