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Chapter 3: Problem 16

Let \(A\) be a \(5 \times 8\) matrix with rank equal to 5 and let b be any vector in \(\mathbb{R}^{5}\). Explain why the system \(A \mathbf{x}=\mathbf{b}\) must have infinitely many solutions.

Short Answer

Expert verified
Matrix A is a \(5 \times 8\) matrix with rank 5, meaning all its rows are linearly independent. As b is any vector in \(\mathbb{R}^{5}\), it can be expressed as a linear combination of rows of A, implying the rank of A equals the rank of the augmented matrix [A | b]. Since the number of variables (8) is greater than the rank of A (5), there must be infinitely many solutions for the linear system \(A \mathbf{x} = \mathbf{b}\).

Step by step solution

01

Establish the dimensions of the matrix and solution space

- Dimensions: Matrix A is a 5 x 8 matrix, which means it has 5 rows and 8 columns. - Solution space: In an equation Ax = b, x is a vector in 鈩漗8, meaning that x is a column vector with 8 elements in it.
02

Analyze the rank of the matrix

The rank of A, given as 5, represents the maximum number of linearly independent rows in A. Since A has 5 rows and rank 5, all of its rows are linearly independent.
03

Relate the rank of A to the augmented matrix [A | b]

The rank of the augmented matrix [A | b] can't be greater than the rank of A, because adding a column (vector b) cannot create new linearly independent rows. There are two cases to consider: 1. If rank([A | b]) = rank(A), there will be at least one solution to the system Ax = b. 2. If rank([A | b]) < rank(A), there will be no solution to the system Ax = b. However, it is given that b is any vector in 鈩漗5, which means that it can always be expressed as a linear combination of the rows of A since all rows of A are linearly independent. Hence, rank(A) = rank([A | b]).
04

Determine the existence of solutions and their nature

Since rank(A) = rank([A | b]), there always exists at least one solution for Ax = b. Now, we need to determine whether the solution is unique or there are infinitely many solutions. The number of solutions for a system Ax = b depends on the number of variables (8, corresponding to 鈩漗8) and the rank of A. In this scenario: - Number of variables (n) = 8 - Rank of A (r) = 5 Since the number of variables is greater than the rank of A (n > r), there are infinitely many solutions for the linear system Ax = b.

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Let \(S\) be the subspace of \(P_{3}\) consisting of all polynomials \(p(x)\) such that \(p(0)=0,\) and let \(T\) be the subspace of all polynomials \(q(x)\) such that \(q(1)=\) 0\. Find bases for (a) \(S\) (b) \(T\) (c) \(S \cap T\)

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