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Chapter 2: Problem 1

Show that the following vectors are linearly independent (over \(\mathbf{C}\) or \(\mathbf{R}\) ). (a) \((1,1,1)\) and \((0,1,-2)\) (b) \((1,0)\) and \((1,1)\) (c) \((-1,1,0)\) and \((0,1,2)\) (d) \((2,-1)\) and \((1,0)\) (e) \((x, 0)\) and \((0,1)\) (f) \((1,2)\) and \((1,3)\) (g) \((1,1,0),(1,1,1)\), and \((0,1,-1)\) (h) \((0,1,1),(0,2,1)\), and \((1,5,3)\)

Short Answer

Expert verified
In summary, for all given sets of vectors in the problem statement, their corresponding matrices have nonzero determinants. Therefore, these sets of vectors are linearly independent: (a) \((1,1,1)\), \((0,1,-2)\) (b) \((1,0)\), \((1,1)\) (c) \((-1,1,0)\), \((0,1,2)\) (d) \((2,-1)\), \((1,0)\) (e) The vectors \((x,0)\) and \((0,1)\) are linearly independent if x is not equal to 0 (f) \((1,2)\), \((1,3)\) (g) \((1,1,0)\), \((1,1,1)\), \((0,1,-1)\) (h) \((0,1,1)\), \((0,2,1)\), \((1,5,3)\)

Step by step solution

01

Case (a)

We are given 3-dimensional vectors \((1,1,1)\) and \((0,1,-2)\). To check if these vectors are linearly independent, form a matrix by putting the given vectors as its columns and find the determinant of the corresponding matrix. If the determinant is nonzero, the vectors are linearly independent. Form the matrix and find its determinant: \[A = \begin{bmatrix} 1 & 0 \\ 1 & 1 \\ 1 & -2 \end{bmatrix}\] The determinant for a 2x2 matrix is calculated as: \[ \det(A) = (1 * 1) - (0 * -2) = 1 \neq 0\] Since the determinant is nonzero, the vectors \((1,1,1)\) and \((0,1,-2)\) are linearly independent.
02

Case (b)

We are given 2-dimensional vectors \((1,0)\) and \((1,1)\). Form the matrix and find its determinant: \[A = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}\] The determinant for a 2x2 matrix is calculated as: \[ \det(A) = (1 * 1) - (1 * 0) = 1 \neq 0\] Since the determinant is nonzero, the vectors \((1,0)\) and \((1,1)\) are linearly independent.
03

Case (c)

We are given 3-dimensional vectors \((-1,1,0)\) and \((0,1,2)\). Form the matrix and find its determinant: \[A = \begin{bmatrix} -1 & 0 \\ 1 & 1 \\ 0 & 2 \end{bmatrix}\] The determinant for a 2x2 matrix is calculated as: \[ \det(A) = (-1 * 2) - (0 * 1) = -2 \neq 0\] Since the determinant is nonzero, the vectors \((-1,1,0)\) and \((0,1,2)\) are linearly independent.
04

Case (d)

We are given 2-dimensional vectors \((2,-1)\) and \((1,0)\). Form the matrix and find its determinant: \[A = \begin{bmatrix} 2 & 1 \\ -1 & 0 \end{bmatrix}\] The determinant for a 2x2 matrix is calculated as: \[ \det(A) = (2 * 0) - (1 * -1) = 1 \neq 0\] Since the determinant is nonzero, the vectors \((2,-1)\) and \((1,0)\) are linearly independent.
05

Case (e)

We are given 2-dimensional vectors \((x,0)\) and \((0,1)\). Form the matrix and find its determinant: \[A = \begin{bmatrix} x & 0 \\ 0 & 1 \end{bmatrix}\] The determinant for a 2x2 matrix is calculated as: \[ \det(A) = (x * 1) - (0 * 0) = x \] Since the determinant is equal to x, the vectors \((x,0)\) and \((0,1)\) are linearly independent if x is not equal to 0.
06

Case (f)

We are given 2-dimensional vectors \((1,2)\) and \((1,3)\). Form the matrix and find its determinant: \[A = \begin{bmatrix} 1 & 1 \\ 2 & 3 \end{bmatrix}\] The determinant for a 2x2 matrix is calculated as: \[ \det(A) = (1 * 3) - (1 * 2) = 1 \neq 0\] Since the determinant is nonzero, the vectors \((1,2)\) and \((1,3)\) are linearly independent.
07

Case (g)

We are given 3-dimensional vectors \((1,1,0)\), \((1,1,1)\), and \((0,1,-1)\). Form the matrix and find its determinant: \[A = \begin{bmatrix} 1 & 1 & 0 \\ 1 & 1 & 1 \\ 0 & 1 & -1 \end{bmatrix}\] The determinant for a 3x3 matrix is calculated as: \[ \det(A) = 1*(-1*-1) - 1*(1*(-1)-1*1)+0 = 2 \neq 0\] Since the determinant is nonzero, the vectors \((1,1,0)\), \((1,1,1)\), and \((0,1,-1)\) are linearly independent.
08

Case (h)

We are given 3-dimensional vectors \((0,1,1)\), \((0,2,1)\), and \((1,5,3)\). Form the matrix and find its determinant: \[A = \begin{bmatrix} 0 & 0 & 1 \\ 1 & 2 & 5 \\ 1 & 1 & 3 \end{bmatrix}\] The determinant for a 3x3 matrix is calculated as: \[ \det(A) = 1*(-1*(1-10)) - 0 - 0 = 9 \neq 0\] Since the determinant is nonzero, the vectors \((0,1,1)\), \((0,2,1)\), and \((1,5,3)\) are linearly independent.

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Key Concepts

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

Determinant
In linear algebra, the determinant is a special number that can be calculated from a square matrix. It provides essential information about the matrix and the linear transformation it represents. The determinant is particularly useful for checking the linear independence of vectors.
  • A 2x2 matrix has a simple formula for its determinant: \(\det(A) = ad - bc\) for matrix \(A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\).
  • If the determinant is nonzero, the matrix is invertible, meaning the corresponding vectors are linearly independent.
  • For a 3x3 matrix, the determinant can be computed using expansion by minors or other methods, which involve more complex calculations.
Understanding determinants is crucial in solving systems of linear equations and finding eigenvalues. In various practical applications, determinants also help evaluate the stability of systems, such as in engineering and physics.
Matrix
A matrix is an essential concept in linear algebra, consisting of a rectangular array of numbers or, more broadly, elements that can represent various mathematical or physical systems.
  • Matrices are not just simple grids; they represent linear transformations and hold the framework for vector spaces.
  • The size of a matrix is given by its dimensions, indicated by rows and columns, such as a 2x2 or 3x3 matrix.
  • Matrices serve multiple purposes, including solving systems of linear equations, performing linear transformations, and representing graphs in computer science.
Matrices can be added, subtracted, and multiplied following specific rules, which makes them very versatile tools in mathematics. Their manipulation forms the groundwork for understanding more complex concepts like eigenvalues and eigenvectors.
Vector Spaces
Vector spaces, or linear spaces, are fundamental structures in linear algebra, offering a setting in which vectors can be added together and multiplied by scalars. This simple yet powerful framework allows extensive exploration of geometric and algebraic properties.
  • Vectors in a vector space can be of any size as long as they comply with the space's rules鈥攁dding vectors and scaling them should still yield vectors in the same space.
  • A vector space must satisfy certain axioms, such as associativity of addition, existence of a zero vector, and distributive properties.
  • Within a vector space, concepts like basis and dimension offer insights into the structure. For instance, the basis of a vector space is a set of linearly independent vectors that span the entire space.
Understanding vector spaces enriches comprehension of dimensions and vectors, which is crucial for fields like quantum mechanics and computer graphics.
Linear Algebra
Linear algebra is the branch of mathematics concerning linear equations, linear functions, and their representations through matrices and vector spaces. It is essential for many areas of mathematics and its applications.
  • Linear algebra involves studying lines, planes, and subspaces, but in higher-dimensional spaces.
  • Core topics in linear algebra include solving linear equations, understanding vector spaces, exploring matrices, and functions of vector-space linearity.
  • The knowledge of linear algebra underpins various fields such as engineering, computer science, physics, economics, and statistics.
By mastering linear algebra, one gains the tools necessary for solving real-world problems, including systems of equations, resource allocation situations, and various optimization problems in machine learning and data science.

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Most popular questions from this chapter

Let \(V=K^{3}\) for some field \(K\). Let \(W\) be the subspace generated by \((1,0,0)\), and let \(U\) be the subspace generated by \((1,1,0)\) and \((0,1,1)\). Show that \(V\) is the direct sum of \(W\) and \(U\).

Find the coordinates of the vector \(X\) with respect to the vectors \(A, B, C\). (a) \(X=(1,0,0), A=(1,1,1), \quad B=(-1,1,0), C=(1,0,-1)\) (b) \(X=(1,1,1), A=(0,1,-1), B=(1,1,0), \quad C=(1,0,2)\) (c) \(X=(0,0,1), A=(1,1,1), \quad B=(-1,1,0), C=(1,0,-1)\)

Let \(V\) be the vector space of continuous functions on the interval \([-\pi, \pi] .\) If \(f, g\) are two continuous funetions on this interval, define their scalar product \(\langle f, g\rangle\) to be $$ \langle f, g\rangle=\int_{-\nabla}^{\pi} f(t) g(t) d t $$

Let \(c\) be a rational number \(>0\), and let \(\gamma\) be a real number such that \(\gamma^{2}=c .\) Show that the set of all numbers which can be written in the form \(a+b \gamma\), where \(a, b\) are rational numbers, is a field.

Show that the following sets of elements in \(\mathbf{R}^{3}\) form subspaces. (a) The set of all \((x, y, z)\) sueh that \(x+y+z=0\). (b) The set of all \((x, y, z)\) such that \(x=y\) and \(2 y=z\) (e) The set of all \((x, y, z)\) such that \(x+y=3 z\).
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