/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 21 Let \(\mathrm{AP}(4,3)\) be the ... [FREE SOLUTION] | 91影视

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Chapter 9: Problem 21

Let \(\mathrm{AP}(4,3)\) be the set of \(4 \times 3\) matrices of the form $$ \left[\begin{array}{ccc} a & a+b & a+2 b \\ a+c & a+c+b & a+c+2 b \\ a+2 c & a+2 c+b & a+2 c+2 b \\ a+3 c & a+3 c+b & a+3 c+2 b \end{array}\right] $$ where \(a, b, c \in \mathbb{R} .\) Prove that \(\operatorname{AP}(4,3)\) is a vector space and find a basis for it.

Short Answer

Expert verified
\(\mathrm{AP}(4,3)\) is a vector space. Its basis is \(\left\{A_1, A_2, A_3\right\}\), where \(A_1, A_2, A_3\) represent matrices formed by variations in \(a, b, c\).

Step by step solution

01

Define the Operation of Addition

To prove that a set is a vector space, we need to show it is closed under addition. Consider two arbitrary matrices from \( \mathrm{AP}(4,3) \), denoted as \( A \) and \( B \), with parameters \( a_1, b_1, c_1 \) for \( A \) and \( a_2, b_2, c_2 \) for \( B \). Use these to describe the sum \( A + B \). Each element of the matrices is the sum of corresponding elements. Therefore, the resulting matrix will have parameters \( (a_1+a_2), (b_1+b_2), (c_1+c_2) \). Thus, it is closed under addition as the sum of any two such matrices is still in the same form.
02

Define the Operation of Scalar Multiplication

We must also show closure under scalar multiplication. Let \( \alpha \) be a scalar in \( \mathbb{R} \) and multiply it by a matrix \( A \) from \( \mathrm{AP}(4,3) \). This operation results in a matrix with parameters \( (\alpha a), (\alpha b), (\alpha c) \). Hence, scalar multiplication maintains the matrix's form in \( \mathrm{AP}(4,3) \), demonstrating closure under scalar multiplication.
03

List Vector Space Axioms

Verify that all vector space axioms are satisfied: 1) Closure under addition and scalar multiplication (shown in steps 1 and 2). 2) Existence of a zero vector (matrix with 0 in all entries). 3) Existence of additive inverses (multiply the matrix by -1). 4) Associativity and commutativity of addition. 5) Distributive and associative properties of scalar multiplication. Since these conditions are satisfied, \( \mathrm{AP}(4,3) \) is a vector space.
04

Find a Basis

To find a basis, look for matrices in \( \mathrm{AP}(4,3) \) corresponding to unit variations of one parameter at a time, like setting other parameters to zero. We generate matrices: 1) Set \( (a,b,c) = (1,0,0) \) for the first basis matrix. 2) Set \( (a,b,c) = (0,1,0) \) for the second basis matrix. 3) Set \( (a,b,c) = (0,0,1) \) for the third basis matrix. These matrices are linearly independent and any matrix in \( \mathrm{AP}(4,3) \) can be expressed as a linear combination of these matrices.

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Key Concepts

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

Linear Combination
In the context of vector spaces, a linear combination refers to the expression of a vector as the sum of scalar multiples of other vectors. For example, in the set \( \mathrm{AP}(4,3) \), any matrix can be written as a linear combination of basis matrices. A basis matrix is a matrix that helps to build up the entire space by combining multiples of its components.
  • Consider three matrices formed by making each parameter \( a, b, \) and \( c \) non-zero one at a time while setting others to zero, like \( (1,0,0), (0,1,0), \) and \( (0,0,1) \).
  • Any matrix in the set can be constructed by a linear combination of these three matrices. This means, for any matrix formed in \( \mathrm{AP}(4,3) \), we can find appropriate multipliers for our basis matrices to express it as a sum of these matrices.
Understanding linear combination allows us to see how versatile these matrices are for constructing any matrix within \( \mathrm{AP}(4,3) \).
Matrix Operations
Matrix operations, particularly addition and scalar multiplication, are crucial in proving whether a set like \( \mathrm{AP}(4,3) \) forms a vector space. Both operations must keep matrices within the defined space.To illustrate:
  • Addition: When you add two matrices from \( \mathrm{AP}(4,3) \), each element of the resulting matrix is the sum of the corresponding elements from the original matrices. If you start with two matrices \( A \) and \( B \), having parameters \( (a_1, b_1, c_1) \) and \( (a_2, b_2, c_2) \), the sum \( A + B \) will have parameters \((a_1+a_2, b_1+b_2, c_1+c_2)\). This verifies that adding two matrices keeps the result in \( \mathrm{AP}(4,3) \).
  • Scalar Multiplication: When a matrix is multiplied by a scalar \( \alpha \), every element inside the matrix is scaled. For a matrix with parameters \( (a, b, c) \), multiplying by \( \alpha \) results in a matrix with parameters \( (\alpha a, \alpha b, \alpha c) \). This operation also keeps the matrix within the set.
These operations are essential in ensuring that \( \mathrm{AP}(4,3) \) meets vector space requirements.
Basis
A basis in a vector space is a set of vectors that are linearly independent and span the entire space. For \( \mathrm{AP}(4,3) \), identifying a basis allows us to describe any matrix in terms of simpler matrices.The process for finding a basis typically includes:
  • Linear Independence: The selected matrices must be linearly independent. This means no matrix can be represented as a linear combination of the others. In \( \mathrm{AP}(4,3) \), the matrices formed by \( (1,0,0), (0,1,0), \) and \( (0,0,1) \) values fit this criterion.
  • Spanning the Space: Any matrix in the space should be expressible as a linear combination of the basis matrices. Our chosen three matrices allow us to write any matrix by adjusting the multiples of these three basis matrices.
Thus, the basis provides a foundation for constructing the entire vector space.
Closure Properties
Closure properties are a cornerstone of vector space theory. For \( \mathrm{AP}(4,3) \), we need to ensure closure under two operations: addition and scalar multiplication.
  • Closure Under Addition: Adding any two matrices should still yield a matrix within the space. As shown, if \( A \) and \( B \) are both in \( \mathrm{AP}(4,3) \), \( A + B \) remains in the same space, showcasing closure under addition.
  • Closure Under Scalar Multiplication: This involves multiplying a matrix by a scalar, and the result should remain within the space. When any matrix from \( \mathrm{AP}(4,3) \) is multiplied by any real number \( \alpha \), it remains within the set of \( \mathrm{AP}(4,3) \).
These closure properties confirm that the set behaves according to vector space principles, supporting the related operations without altering the set's nature.

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Most popular questions from this chapter

Show that there are, in \(\mathbb{R}^{4}\), infinitely many different subspaces of dimension \(1 .\) Are there, likewise, infinitely many of dimension (a) \(2 ;\) (b) \(3 ;\) (c) 4; (d) \(0 ?\)

What is the dimension of the subspace of \(M_{n}(\mathbb{R})\) comprising: (i) all upper triangular matrices; (ii) all skew-symmetric matrices?

Determine whether or not the following sets are bases for the vector spaces indicated; (i) For \(\mathbb{R}^{3}\) (a) \(\\{(1,2,3),(4,5,6),(7,8,9)\\} ;\) (b) \(\\{(1,2,3),(6,5,4),(7,8,9)\\}\) (c) \(\\{(1,0,3),(0,5,0),(7,0,9),(11,13,15)\\} ;\) (d) \(\\{(13,12,11),(10,9,8)\\}\) (e) \(\\{(\sqrt{1}, \sqrt{2}, \sqrt{3}),(\sqrt{4}, \sqrt{5}, \sqrt{6}),(\sqrt{7}, \sqrt{8}, \sqrt{9})\\}\). [(c) and (d) can be done much more easily after you have read Corollary 2.] (ii) For \(\mathbb{R}^{4}:\) (a) \(\\{(2,1,7,7),(-5,-3,53,48),(-1,0,2,3),(1,2,10,15)\\}\) (b) \(\\{(1,1,1,1),(1,1,1,0),(1,1,0,0),(1,0,0,0)\\}\) (c) \(\\{(3,1,4,1),(5,9,2,6),(5,3,5,8),(9,7,9,3)\\}\) (d) \(\\{(1,1,-1,2),(2,1,1,-1,),(-1,2,1,1),(1,-1,2,1)\), (e) \(\\{(2,-3,4),(-5,6,-7),(1,1,1)\\}\)

Find a basis for each of the following subspaces: (i) \(\\{(x, y, z): 4 x+5 y-z=0\\}\) in \(\mathbb{R}^{3}\); (ii) \(\\{(x, y, z, t): x+3 y=5 z+7 t\\}\) in \(\mathbb{R}^{4}\); (iii) \(\\{(x, y, z, w, t): 3 x+y+4 z+w+5 t=9 x+2 y+6 z+5 w+3 t=3 x-2 z+3 w-7 t=0\\}\) in \(\mathbb{R}^{5}\);(iv) \(\\{p(x): p(3)=0\\}\) in \(\mathbb{R}_{4}[x] ;(\mathrm{v})\\{p(x): p(3)=p(1)=0\\}\) in \(\mathbb{R}_{6}[x]\)

Find a basis for the subspace \(\left\\{a_{0}+a_{1} x+a_{2} x^{2}+a_{3} x^{3}: a_{0}+a_{1}+a_{2}+a_{3}=0\right\\}\) of \(\mathbb{R}_{3}[x]\).
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