/*! 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 32 Suppose \(A\) is an \(m\) -by- \... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 32

Suppose \(A\) is an \(m\) -by- \(n\) matrix of real numbers. Prove that the dimension of the span of the columns of \(A\) (in \(\mathbf{R}^{m}\) ) equals the dimension of the span of the rows of \(A\) (in \(\mathbf{R}^{n}\) ).

Short Answer

Expert verified
To prove that the dimension of the column space of matrix A equals the dimension of the row space, we can use the rank of the matrix. Perform row reduction to reach the row-echelon form (REF) of matrix A, and calculate the rank of matrix A. Then, perform column reduction to achieve column-echelon form (CEF) and calculate the rank again, noticing it remains constant. This constant rank represents the linearly independent columns and rows in matrix A. By demonstrating that the rank (linearly independent rows/columns) remains the same after row and column operations, we can conclude that the dimension of the column space (in R^m) equals the dimension of the row space (in R^n).

Step by step solution

01

Understand Column Space and Row Space

Column space (C(A)) and row space (R(A)) are two important vector spaces of a matrix A. Column space (C(A)) is the span of all linear combinations of columns of A, it is a subspace of R^m. Row space (R(A)) is the span of all linear combinations of rows of A, it is a subspace of R^n. So for matrix A (m x n), the column space will be a subspace in R^m and the row space will be a subspace in R^n.
02

Perform Row Reduction of Matrix A

Taking the given matrix A, perform row reduction or Gaussian elimination to reach the matrix into its row-echelon form (REF). In the process of row reduction, one should keep track of the elementary row operations performed on matrix A.
03

Determine the Rank of Matrix A

The rank of a matrix can be determined by counting the number of non-zero rows in its row-echelon form. Let's denote the rank of matrix A as 'rank(A)'. Rows that aren't non-zero will be called free rows.
04

Find the Column Space Dimension

To find the dimension of the column space of matrix A, count the number of linearly independent columns. It is equivalent to the rank of the matrix, i.e., the dimension of the column space of matrix A is 'rank(A)'.
05

Perform Column Reduction of Matrix A

We will perform column reduction (similar to row reduction) to achieve column-echelon form (CEF) of matrix A. It is crucial to ensure that the number of linearly independent columns remains the same after column operations in matrix A.
06

Determine the Rank of Matrix A

After performing column reduction, determine the rank of matrix A by counting the number of non-zero columns in its column-echelon form. The number of linearly independent rows will stay the same, which means 'rank(A)' will not change.
07

Find the Row Space Dimension

To find the dimension of the row space of matrix A, count the number of linearly independent rows. It is equivalent to the rank of the matrix, i.e., the dimension of the row space of matrix A is 'rank(A)'.
08

Prove the Dimensions are Equal

After performing the steps for finding the dimensions of the column and row spaces, we can see that the rank of matrix A (linearly independent rows or columns) has remained constant throughout the process. Thus, the dimension of the column space (in R^m) is equal to the dimension of the row space (in R^n). From the results of Steps 4 and 7, we conclude that the dimension of the span of the columns of matrix A (in R^m) equals the dimension of the span of the rows of matrix A (in R^n).

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