The Importance of Matrix Span and Matrix Multiplication in Linear Algebra | Analyzing the Span of the Columns of A^5

If A is a 5 × 5 matrix whose columns span R5, then the columns of A^5 also span R5. (T/F)

The statement is true

The statement is true.

Let’s first understand some key concepts related to the question:

1. Span: The span of a set of vectors is the set of all possible linear combinations of those vectors. In other words, if a set of vectors spans a certain vector space, it means that any vector in that space can be expressed as a linear combination of those vectors.

2. Matrix: A matrix is a rectangular array of numbers arranged in rows and columns.

3. Matrix Multiplication: Matrix multiplication involves multiplying the elements of rows of one matrix with the corresponding elements of the columns of another matrix. The resulting matrix is obtained by summing the products.

Now, let’s analyze the given statement. We have a 5 × 5 matrix A, with columns that span R5. This means that any vector in R5 can be expressed as a linear combination of the columns of A.

We need to determine if the columns of A^5, which represents the matrix A multiplied by itself five times, still span R5.

Consider the dot product of A with itself, i.e., A · A. This dot product will have the same number of columns as the original matrix A, which is 5. Each column of A · A will be a linear combination of the columns of A.

Repeating this process five times, A^5 = A · A · A · A · A, we can see that each column of A^5 is a linear combination of the columns of A. Since the original columns of A span R5, it follows that the columns of A^5 also span R5.

Therefore, the statement is true.

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